There are several key points, of which I will pick my top ones.

1. The human vision system will re-focus very quickly, and only at what it is looking at in that moment. It is therefore difficult to look at a scene and see any kind of out of focus blur. This will cause humans to not realize the effect of certain areas being out of focus — you can't trust your eyes, unless you train very carefully for it.
2. The human vision system quickly adjusts for contrast, making it difficult to see just how contrasty a particular scene is. Thus you can't trust the un-trained eye to get the right shot. It takes practice and skill to get the right dynamic range translated.
3. We see in 3 dimensions, which can sometimes be difficult to translate into 2.
4. Even though we have a uniform perceived field, actually the resolution in the central part of our vision is much higher than the edges. Also, the edge part of our vision is better at dark perception than the central part. Also, we tend to ignore things that aren't our key point of interest. What all of this translates to, when we are taking a picture: we don't see the trash can in the scene, or the pole sticking out of someone's head, unless we specifically look for them.
5. We don't actually create a snapshot at any one time — we operate something like a movie camera, but each "pixel" operates independently. Still, our mind is able to make it look like we have a snapshot. It's therefore kind of hard for us to really understand shutter speed.