Makes a straight surface where none presently exists. Has nothing to do with thickness. This tool creates the first surface from which all other operations are referenced. It is used as the first step in the milling process.
Makes a parallel surface to a given reference surface. Has nothing to do with straightness. If your reference surface is straight, the surface the planer creates will be straight as well. If you put in a banana, you'll get a slightly thinner banana out.
Those are the very basic descriptions. You can get a planer to do some of what a jointer does if you provide workpiece support (sleds, wedges, etc). It's time consuming but possible. These two tools work hand-in-hand when milling rough stock. I use the following method with rough stock:
1. Cut to rough length for a given workpiece.
2. Joint one face on the jointer (not edge).
3. Plane the opposite face with the planer.
4. Joint one edge on the jointer.
5. Rip to width and/or plane to width if possible.
Many folks joint both the face and the edge without going to the planer. I dislike this method because i feel it's limiting. I get both faces flat and parallel so that I will have 4 choices when jointing an edge given the direction of the grain.