The story was originally supposed to be a oneshot, but as I tried to explain my interpretation of boggarts I came up with more examples.
My version of Neville in this is particularly sensitive to magic in living things, hence his skill at Herbology (I imagine he'd also be quite good at healing). It is not mentioned in canon where Neville is during the attack on his parents, but it is quite reasonable to assume that he was hidden somewhere nearby and able to both hear his parent's torture, and feel the associated Dark magic / Dark Marks.
Neville taps his quill on the blank sheet of parchment, trying to plan his DADA essay. The lesson on boggarts was at once both one of the best and worst he has ever had. First Snape insulted him – again – in front of the only teacher who did not yet know what a failure he is, yet then Professor Lupin stood up for him. He had to show the entire class that the thing he fears most is the greasy Potions professor, but then his spell actually worked and for the first time they were laughing with him not at him. The next Potions class was absolute torture, but as the story spread he found himself almost popular.
He sighs and turns his attention back to the assignment. As crazy as the last week has been, he knows he is just putting off having to think about what the boggart really meant. While the rest of the school is enjoying his retaliation against his Snape-boggart, he knows that Professor Lupin was right, the professor is not his true fear.
He can't quite explain the feeling he gets from Professor Snape, although the boggart made it stronger and clearer. It is not just the man's intimidating presence, or his constant derision, there is something more.
There is a strange aura about the professor, one he feels any time he gets too close, the same aura echoing even more strongly from his boggart form. It has always been dark and slimy, yet when the boggart approached, for the first time he heard screams. Two voices, one male and one female, a piercing duet of pure agony.
He has never asked his Grandmother about that night, never dared to, but he knows what those voices are. His father and mother, screaming under the Crutiatus as they are tortured into insanity.
He does not know quite how that night links to his school professor or his greatest fear, but thinking about it, he can start piecing the answers together. The dark aura is unusual, but he recognises it now as his sense of magic, dark magic, the same magic which permeated the room where his parents lost themselves. It should surprise him to sense the aura clinging to one of his professors, but it doesn't. He has heard the rumours from his grandmother's acquaintances and the other students, and it simply appears that there is more truth than he thought. The man himself doesn't do much to counteract them.
Once that link is understood, defining his fear is quite simple. Hogwarts is where he is supposed to be safe. But if Snape is there, with that aura, what is to stop the rest from playing out. Every touch of the dark magic reminds him that next time it could be him lying in that ward at St Mungo's, or his friends, or anyone really. It reminds him that insanity is only ever a curse away.