"A bit" vs. "a little bit" vs. "a little"

There is no difference in meaning between these three sentences, unless the speaker already establishes a context.

He's a bit angry

This means he is somewhat angry.

He's a little bit angry

This means he is somewhat angry. No native speakers would infer any qualitative difference in how angry he is, unless someone first said

John: He's a bit angry (meaning fairly angry)

Mary: No, he's only a little bit angry (meaning not as angry as John implied)

When used alone, "little", "bit", and "little bit" all mean "small". But if you emphasize "little bit" over "little" or "bit" then you are emphasizing how small it is.

If I walked up to you on Monday and said "I'm a bit hungry", and on Tuesday I walked up to you and said "I'm a little bit hungry", there is no way decide that I'm not as hungry Tuesday as I was Monday. But if on Wednesday I said "I'm a bit hungry", and you said "Have a hamburger", I could clarify that I'm not as hungry as you think, by saying "I'm only a little bit hungry".

They don't quite mean the same thing. It's subtle, though.

He is a bit angry, but mostly he's just glad you're back.

Using a "bit angry" would seem to indicate that he's something else, too, as "bit" indicates only one part of the whole.

He is a little angry and would appreciate an apology.

Using a "little angry" is slightly more formal. Also contrast with the above.

He is a little angry right now. Please don't antagonise him any more.

In this case, the subject's mindset is all a little angry, as if it were slowly suffusing him.

He is a little bit angry. Don't worry about it.

By using two diminutive words together, it minimises the degree of anger. I imagine this being used either as a reassuring response to a question, or as sarcasm.

Is he angry with me?
Only a little bit. He'll calm down soon.

Was he angry when he found out I'd written off the Porsche?
He was a little bit angry, yes.