How would this have gone differently if Snape had given Harry a hug instead?

I briefly considered doing that (but still having Harry, after the “Professor” line, reject Snape's affection and run away). But, no, Harry's world is a world of very little affection. There's an interesting compromise going on here, between the atmosphere of the world and the action of each page: Setting the atmosphere is very important, but when characters being emotionally distant happens so often, it doesn't have any special impact when it does happen.

On another note, Voldemort has some very cool lines in the Harry Potter series. “Voldemort is my past, present and future” is something that ze (as the memory of Tom Riddle) says in the second book. One of the coolest things about “alternate universe” fanfiction in general is when you can take an event from the original story, but show it in a different context and with different reasons behind it.