There are some pieces of headgear for which the in-game description and/or the description you find online is either inaccurate or incomplete. Here are a few notes on such items that should help:

Goggles: The healing from water is rather slow; the maid cap is probably better when maxed.
Cat Ears: This does not reveal the number of chests/jars in the area; rather, it reveals them on the mini-map.
Vampire Kit: This will only heal you if you actually hit an enemy (and I think it is necessary to do damage); hitting a pot or rock will not count.
Monkey Hat: The extra money is checked when you kill the enemy or break the jar/rock; it does not matter what you are wearing when you pick up the money. (Hint: If you see a piggy bank, equip this hat before breaking it.)
Nightcap: This does not increase critical damage, nor does it make criticals more common. Rather, it increase the rate at which the critical count (at the top of the screen) increases, so that you only need 50 or 34 criticals to gain a level of drill power.
Samurai Helm: No falling damage immunity until level 5. Level 3 gives you water immunity, level 4 gas immunity, and level 5 trap and falling damage immunity. Also, note that falling damage only happens when you fall out-of-bounds in this game; falling onto solid ground never damages you.
Magic Hood: The elemental damage bonus stacks with the usual attack damage bonus; al level 5 you will do 4.5x damage with special attacks than you would without any headgear. Note that the bonus only applies to special attacks, and I think the one you use on the ground without the spin dash is not affected (though I think the one you get by attacking afterwords is).

There's one other thing worth mentioning:

The Headband, Helmet, and Magic Hood are found in Mystery Bags. Which one has which hat is randomly determined when you start a new game. This means that, while you could get lucky and get one of them early (on my first play (Normal) I got the Headband before the first boss), there is no guarantee you will get them until you've found all the Mystery Bags, and you might get unlucky and not get any of them until the very end of the game. Therefore, when preparing for New Game +, I recommend assuming that you won't get them early; you are better off leveling up something else (like the Vampire Kit) to level 3 if you want to have an early game advantage.

(Of course, you can still level these up later; just don't rely on being able to find them; also, these three headgears can be useful for the game's superboss, and for boss rush, where you are guaranteed to have them if you get all of the Mystery Bags.)