For example, if I want to say "the performance of this bike could be affected/influenced by many things", is there a difference between these two words?
My gut feeling is that "affect" sounds better, but I cannot find evidence in dictionaries.
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. (We do speak of mechanistic processes being influenced, though, especially complex ones.)
I feel like "affected" as a slightly negative connotation whereas "influence" can go either way.
In the following sentence (made up for the occasion) ...
"The hand of a magnet takes its position under the influence of the Earth magnetic field but it can also be affected by the presence a nearby metallic object."
... one can clearly see the difference.
You will find that affect has several other negative meanings:
In the example above "influence" is neutral but it can actually go either way. It can be deemed counter productive or beneficial. However only your own judgement will tell you; depending on the nature of the influence.
In other words your opinion will not be influenced by the fact that "influence" is usually positive or usually negative ;-)
Arguably though, "influential" has a slightly positive orientation.
Influenced means "having the character, development, or behavior changed by something or somebody."
Affected means "influenced or touched by an external factor."
Affected doesn't seem to have a negative connotation (and in most the cases it is considered synonym of influenced), except in sentences like "the gesture appeared both affected and stagy." In such sentences, affected means "artificial, pretentious, and designed to impress."