DISCLAIMER: I do not own God of War or Devil May Cry.

The last demon fell onto its back, screaming as Kratos picked it up, holding a blade in front of its face.
"Tell me where I can find him!"
The terrified demon pointed down the street, and Kratos dropped it, walking in the direction it had pointed. He soon found the building he was looking for. It was a small building with a big reddish pink sign in the front. Kratos walked over to the door and violently kicked it open, the door flying off its hinges, into the shop, and smashing to pieces on the rear wall, leaving a deep dent in the stone.
A man with white hair, wearing a red coat, stepped out of the back doorway, looking at Kratos with a certain level of interest.
"That door's never locked, dude."

Kratos approached, gripping Dante by the collar and glaring at him.
"I need one of your weapons! Where is it?"
Dante glared back at Kratos.
"So what if I don't tell you, baldy?"
Kratos readied the Blades of Exile, pressing them against Dante's neck. Dante's eyes didn't move, still glaring at Kratos.
"So that's how it is, is it?"

Dante suddenly punched Kratos, staggering him, and kicked him in the chest, launching him back. Kratos landed on his feet, slamming his fist on the floor and causing the entire building to shudder.
"I will kill you!"
Kratos charged, swinging the Blades of Exile at Dante, while Dante dodged the blades, before finally Dante drew Ebony and fired a string of shots at Kratos, watching with interest as Kratos dodged the bullets. Seeing that his opponent was no joke, Dante held out his left hand, and Rebellion flew from its place next to his desk and landed in his hand. Kratos saw the sword come into play, and responded by readying the Blade of Olympus.

Kratos charged with the speed of the Boots of Hermes, and found Dante matching his speed, the two of them exchanging fast, powerful blows with their blades. Dante dived back and fired another bullet, but Kratos blocked the bullet with the Golden Fleece on his shoulder, charging again with the Boots of Hermes. The two continued trading blows, the force of the strikes shaking the buildings around them and filling the walls of the Devil May Cry with cracks. Kratos jumped back and readied the Nemean Cestus, stamping his foot on the ground and once again shaking the entire block, a quake which would one day be traced back to Dante's shop by the state's seismographic institute.
"Show me what you've got!"
Realising what Kratos wanted, Dante summoned the Gilgamesh gauntlet and greave set, assembling himself into a stance.

The two warriors prepared to fight again, a stark contrast to each other. Next to the crushed couch was Kratos, standing in a beastlike stance with the large Cestus on his hands, grinding his teeth in anger as he prepared to attack. Next to the jukebox was Dante, standing in a smooth martial arts stance with the much more compact gauntlets and greaves, leering smugly at Kratos and likely trying to provoke the Spartan. The two both charged, Dante instantly reaching Kratos and swinging his foot at Kratos' head, as Kratos blocked the kick with one of the Cestus. Kratos swung the other Cesti at Dante's jaw, but Dante dived back, landing smoothly on his feet, before launching himself forward again and punching Kratos in the face and launching him out the back of the shop, straight through the wall.

Kratos landed on his feet on the other side of the street, sliding to a stop. Kratos quickly dropped the Nemean Cestus, readying the Barbarian Hammer. Dante came flying from the shop, slamming his fist into the shaft of the Hammer, Kratos blocking the strike with pure strength. As the sheer force of Dante's literally lightning fast strike slammed into the immovable object that was Kratos' massive strength, the building behind Kratos exploded, pieces of it flying into the city and tearing apart yet more buildings. Dante flinched, and Kratos suddenly swung the Hammer, hitting Dante with it and sending him flying across the street, sliding across the ground and ripping up a trail in the concrete. Kratos ran after Dante and punched him with his fist, sending the demon up into the air. Kratos then jumped up and caught Dante in his hands, sticking the Blades of Exile into him and ripping them out brutally, sending blood spraying in all directions.

Kratos then punched Dante down to the ground, the demon's impact with the ground leaving a crater the size of a house in the street and shaking the buildings in the immediate vicinity. Kratos dropped onto the ground just out of the crater, glaring triumphantly at his foe. As he turned to head back to the Devil May Cry, a rock suddenly hit the back of his head, breaking apart on his head. Kratos turned to see Dante standing again, smirking at Kratos. Kratos quickly readied the Bow of Apollo, shooting a barrage of arrows at high speed at Dante, who responded by shooting the arrows out of the air. The exchange of projectiles lasted almost a minute before Dante vanished from the crater, appearing above Kratos and swinging Rebellion down at him. The shockwave from the swing tore up the street down its entire length, culminating with the building at the end of the street splitting apart.

Dante dropped back onto his feet, looking impressed as he regarded Kratos, who still stood where he had been, having blocked that strike with the Blade of Olympus. Kratos cracked his neck, stepping forward and not looking as the ground he had been standing on crumbled. Dante nodded, impressed.
"I haven't met a whole lot of demons who could take that. The Saviour, Abigail, Nero, but that's about it. Except that asshole Mundus, of course, and there's not much that guy can't survive."

Kratos stepped right up close to Dante, glaring at him.
"You aren't human, and the only Olympian I've met who could survive what I did to you is Zeus. Who or what are you?"
Dante shrugged. "If you don't even know who I am, why the hell are you here to kill me? Did Lady or Trish put you up to this? Because trashing my shop and punching me full of holes is kind of everyone's hobby."
Kratos stepped back, folding his arms.
"I was told by Athena that you had a weapon which could rival the Blade of Olympus. Your ultimate sword. I came for it."

Dante stepped back also, growing serious as he heard this.
"That sword… Is dangerous. It's the most dangerous sword in the world. Sorry, but I can't let you have it."
Kratos ground his teeth. "I'll take it!"
Kratos suddenly locked the Claws of Hades onto Dante's chest, attempting to rip his soul from his body. Dante groaned as Kratos struggled to rip his soul out, a struggle which lasted several seconds and sent cracks through the ground around them. Finally, Dante lit up with a burst of red light, sending the Claws flying back into Kratos' hands.

Dante was now in the form of a winged red devil, and flew at Kratos at speeds even Kratos couldn't follow, slamming into him and flying up into the air, the entire city shuddering from the shockwave of Dante's speed. Dante and Kratos grappled in the air, Kratos attempting to overpower Dante, while Dante landed a barrage of energised punches on Kratos, until Kratos finally freed himself from Dante's grip and dropped back to the ground, landing with such impact as to put a crack in the ground that ran the full length of the city. Dante hovered back to the ground, folding his arms and leering down at Kratos.

After a brief staring contest, Dante reverted to his human form, dropping to the ground. Kratos lunged with the Blade of Olympus, while Dante fought back with Rebellion. The two swordsmen, each literally the deadliest and most capable warrior of his time, faced off with their blades once again, sending shockwaves through the entire city with their attacks and toppling buildings for kilometres around. Both were taking minor cuts, but Kratos was taking more of them, and Dante's were vanishing as they were inflicted. Realising this clash wasn't favouring him, Kratos jumped back, but found Dante following him closely, still attacking. Kratos continued to block the barrage of lightning fast strikes, now struggling to keep pace.

As the blades continued striking together, Dante suddenly transformed into his demon form again, his speed massively increasing. Dante gripped Kratos and flew up, the two of them trading high speed blows with the swords still, until Dante deliberately dropped his sword, slamming into Kratos and flying into the sky with him. When the two of them were just below the atmosphere, Dante let go of Kratos and flew back, launching himself into Kratos at positively massive speed, sending Kratos hurtling down to the ground, covered with bruises and cuts, most of the bones in his body broken or at least cracked. Dante flew down and caught Kratos, flying him down to the ground and lowering him onto the concrete, reverting to his human form.

Dante then turned to regard his wrecked shop, and the completely levelled buildings around it, some as far away as several kilometres. He then turned to regard Kratos, who was conscious and trying to stand despite his broken bones, but seemed unable to support his own weight.
"What a mess. I mean the city, not you. Don't suppose you got any third party insurance?"
Kratos finally stood, but was sagging under his own weight.
"I really don't like you."
Dante smirked. "Mutual, baldy. We done here?"
Kratos ground his teeth. He had heard that this weapon was powerful enough to kill Zeus, but he now knew for himself that Dante was deadlier than Zeus. Kratos suspected his mission to take something from Dante had been a suicide mission.
"We're done. I'll leave."
As Kratos limped back towards the portal Athena had used to send him to twenty-first century America, Dante headed back to the Devil May Cry and sat in his chair, admiring his destroyed shop.
"I didn't like the old decorations all that much anyway."

Analysis:

Both sides have advantages, but Dante has most of them.

Strength: While an argument could be made for Kratos having fallen between the cracks in Kronus' hands, obviously that was intended as a showing of Kratos' strength, so obviously it should be counted. While the Saviour's punch was faster than Kronus' strike and had more leverage, Kronus was larger and the differences go in Kratos' favour. Kratos is stronger, even if only by a bit.

Speed: Kratos has feats where he's so fast falling pillars seem to freeze, he can slow time with a statue of the Fates nearby, he has lightning timing reactions as he once reacted to Zeus' lightning, and with the Boots of Hermes we can call his speed supersonic at least, as Hermes can scale Olympus in less than a minute and Kratos gets close to hitting a sonic boom in one of the flying scenes.

Dante hits re-entry friction while running in DMC3, instantly making him at least Mach 10 in his second weakest incarnation. He can also block Beowulf the Lightbeast's blasts of light and Damned Rooks' lasers, and Vergil can do the same with Artemis' lasers, meaning Dante and Vergil have light speed reactions. In DMC1 he can use Griffon's cloud to ground lightning as a set of steps, and is clearly faster than it by virtue of that. Dante can also slow time with Quicksilver, stop it briefly with the Bangle of Time, and is also immune to time control as Nelo Angelo and various weaker demons are. To top it off Dante displays an ability to traverse interstellar distances while in either Devil Trigger or Majin form, as he and Mundus both fly up towards distance stars, and those stars are gone a second later when Dante and Mundus stop.

In short, Kratos is supersonic with the Boots of Hermes, has massively hypersonic reactions, and can slow time to about a tenth making him Mach 10 with sub-relativistic reactions with all his cards on the table at once. Dante is Mach 10 with light speed reactions in his weakest state and can also slow down time to about a 1000th, making Dante's weakest incarnation, with all cards on the table, sub-relativistic with 1000 times FTL reactions, controlling two bodies. In his base form in his normal incarnation he is relativistic with Quicksilver and can also stop time, and in Devil Trigger he is massively faster than light and can still slow or stop time.

Dante is much faster than Kratos.

Surviving the other's attacks:

Dante's attacks against Kratos:

Dante's strikes: Kratos can take some very solid hits. Dante is faster than anything in God of War, and thus will hit with speed that can cut, which may kill Kratos, but in sheer strength Kratos can deal.

Rebellion: Kratos died when he was impaled with a log in God of War, died when impaled with the Blade of Olympus in God of War 2, and likely died again when impaled with the Blade in God of War 3. Hermes died from Kratos cutting his legs off, Zeus was notably damaged when Kratos impaled him, Athena died when Kratos impaled her, Hephaestus died when Kratos impaled him, and Ares died when Kratos impaled him. The list of gods who died from simple blade wounds in God of War is rather long. Dante's Rebellion is also the only object that can take full powered slashes from Yamato, which cuts through reality.

Ebony and Ivory: Kratos has never even seen a gun before, and chances are it won't be easy for him to deal with. There is no reason to believe he is bulletproof, though he could dodge bullets.

Majin form: The Majin form was a match for Mundus who created a pocket dimension containing thousands of visible stars, and was able to damage him even though he later survived that same dimension caving in with him inside it. Kratos has never faced a power of that magnitude.

Kratos' attacks against Dante:

Kratos' strikes: Dante took punches from the skyscraper-sized Saviour and from Abigail who was stronger and much faster than the Saviour. He laughed at both of them.

Blades of Exile: Dante stays whole after being cut in half, literally laughs at impalement, has bulletproof bones and is fast enough to dodge these blades.

Blade of Olympus: Dante has been hit with energy spears from Mundus, which seemed to be destroying life force but at the very least were devastating energy attacks as they noticeably hurt his Majin form. Being impaled is nothing new to Dante, and really the Blade of Olympus is the only weapon Kratos has that may be able to kill Dante.

Claws of Hades: Dante carries for several minutes a haywire neo generator which drains any souls it touches, takes soul attacks from Mundus, and from demons literally called Soul Eaters. He can manipulate souls himself, and since Baul's plan was to take Dante's soul we can safely say that many Devil May Cry demons can mess with souls. Safe to say Dante can resist the Soul Rip.

In the end, Dante has most of the advantage. Kratos is a tough opponent but Dante is just too hard to kill and has too many abilities.