He is a extremelybsweet and smart guy.

We're in the same band together, so I can understand there being slight hesitation, but we have been flirting for 2 MONTHS! He gives me his jacket when I'm cold, he texts me every night, and we hang out frequently.

I know he likes me & he knows I like him, so why won't he ask me out?

You question is lacking information.
We don't even know how old each of you is. High school or college?
Not that that is going to help much. If you don't know, we can only guess.
He could be shy, he could be unsure, he could be not wanting a girlfriend, he could have someone back home, he could be gay (yes flirting and all), he could be too busy, he could be waiting for you to ask, the list goes on.

I guess why have you not asked him out ?

He probably hasn't asked you out yet because he's afraid. Emotions & liking someone puts all of yourself on the line, some people are afraid of that and it's really hard not to be afraid.

He's probably taking it slow, so things take it's natural course rather than rushing it. He's making sure that nothing ruins what you have.

Also he might not be ready to. There might be other factors in his life that he has to take care of first before he adds on a girlfriend.

Don't worry if he really likes you and the timing is right he will ask you out. And remember you can be with someone even without the title.