"Influence" is often used a bit more subtly and often with people: I influenced his decision to go to college. She influenced the board to get her friend a promotion. "Affect" -- at least in my experience -- is more often used to describe a change: I wonder if this vinegar will have any affect on this baking soda? Global warming seems to have a strong affect on climate change.

'Influence" has two main connotations that 'affect' doesn't. The first is that of relation to process rather than state, so that something might affect a rock but rarely would circumstances arise where we could speak of that rock being influenced. The second, and weaker, connotation is of emotive response being involved; this is probably the connotation that makes it feel less appropriate to use in reference to a bike's performance.