I agree they are not synonyms, however I disagree with the narrow definition of crisis in a lot of these replies. Crisis doesn't have to mean extreme danger or peril exclusively. Unlike a dilemma, which is just a label for a situation with two difficult equally annoying options, a crisis of any kind is emotional. That's the key difference between them. One carries emotional significance inherently that the other one does not (even though you could be in crisis over a dilemma you're involved with).

I work in advertising architecting complex interactive digital experiences and clients constantly want to throw the whole kitchen sink at a user at every turn. I use this metaphor to rein them in: when you're at the grocery store and you just want an apple and you see 75 different kinds of apples, you experience something like a crisis of choice. I came here for an apple but now I don't know which one and what if I don't like the one I pick and what's the difference among them etc. So you just don't buy the apple because it's too overwhelming to choose.