All caps are hats, but not all hats are caps! Usually cap refers to a baseball cap, whereas a hat can be a top hat, a bowler, a sombrero, a baseball cap, or any other style you can think of.

The verb 'don' isn't the same as 'wear'. It is an old-fashioned word for the action 'put on'. You may see 'don', and its even more antiquated opposite 'doff', in texts written a hundred years or so ago. The modern equivalents are 'put on' and 'take off'. Be prepared to understand these words if you come across them - but don't use them.

It depends on the shape of the hat. I usually just go by my feel- if the headgear has a small brim in comparison to its body, then I would call it a cap. If it has a wide brim (think something like a cowboy's hat) then I would call it a hat. There could be local differences, but the general rule I just mentioned should be fine and natural in most cases.

Also, I see more "don a hat" in written articles (for the sake of linguistic variety in a simple action). For most cases, I will just say "wear a hat". I almost never see/hear anyone "don a cap". When do I say "don a hat"? Maybe if I saw the pope do it.. but don't count on me saying this very often!