Disclaimer: I don't own Inuyasha or Sesshoumaru. Sadly.
One covets what the other has, and the other only wants to be accepted for what he is. A inu taiyoukai and a inu hanyou; two half-brothers that couldn't differ more.
Sesshoumaru, the older of the two, covets Inuyasha's Tessaiga, and wonders why his father did not leave it to him. Lord Inu no Taisho however, made sure that the barrier around Tessaiga reflected Sesshoumaru, therefore making it only possible for Inuyasha to use it. The Tessaiga, being the Sword of Earth and Protector of Men, has the ability to slay a hundred demons in one swing. This would enable one to have immense power, which is the reason that Sesshoumaru wishes to have it. He had always wanted to surpass his father in strength and power, wishing to prove himself worthy of being his father's son. Exactly why he desires power, we may never know, but in his words, it is because, "The path I walk is the way of Supreme Conquest, and it is power that will reveal the way for me."
Inuyasha is the youngest of the two brothers, a mere hanyou. His mother, Izayoi was a human, and he and Sesshoumaru's father was a full inu taiyoukai. Feeling insignificant compared to his father, Inuyasha seeks the Shikon no Tama, the Jewel of Foul Souls that will allow him to make a wish upon it and become a full demon. This will allow him to become as powerful, if not more powerful than his father, which is all that he really wants.
As a child, Inuyasha was rejected by the humans for his youkai blood, despite his being the son of a human princess. The youkai refused to accept him for what he was due to his human blood. To them, humans are weak and stupid, bearing no purpose in this world. Inuyasha seems to have inherited his mother's kindness, yet at times he can be stubborn and idiotic. What enables Inuyasha to use to the Tessaiga is his compassion for humans; it cannot be used otherwise, unless it is Sesshoumaru. He may use it without the desire to protect humans, only because he is so powerful, but his hand will be burnt afterwards.
Whether Sesshoumaru is evil or not is debatable; but then why does he care for Rin? Why has he softened so much after meeting her? Perhaps it is because, being hurt and vulnerable when he met her, he feels as though he owes her for taking care of him. Rin seems to be the only one that can read Sesshoumaru's emotions, knowing what he's feeling when he doesn't show it or tell anyone. The relationship between these two is still hazy, but one might say she is like a daughter to him. Rin has also been called his ward, for he feels the need to protect her.
Inuyasha, being the main character, could be depicted as one of the "good" characters, yet sometimes all he cares about is himself. Kagome, the re-incarnation of a dead priestess, Kikyo, travels with Inuyasha along with Sango, a youkai taijiya, Miroku, a somewhat lecherous monk, Shippo, a kitsune youkai, and Kirara, Sango's pat and fighting companion. Kagome seems to be the closest to Inuyasha; they even seem to have a love for each other, but neither one will admit it.
So in the end, Sesshoumaru and Inuyasha do have more in common than they both thought. Whether they will admit it or not, both share an affinity for humans, even if the ways differ. No matter what, they are brother, and neither will ever be able to kill the other.