What do you call this hair accessory?

[Trills]

It's a long elastic loop, that goes all the way around your head, with something decorative on it (the decorative thing doesn't go all the way around).

An elastic head band?

Head band

Elastic headband fascinator?

Alice band?

How do I distinguish between one like this that's thin and has a thingy on vs this sporty one?

An Alice band is a hard U-shaped or horseshoe-shaped (ish) one, in my book.

Claire's accessories call them headwraps.

If I heard the word "headwrap" I wouldn't think of that.

I was going to try to buy one online (alternative to fascinator for a wedding) but maybe I should just go into some shops instead.