What's the difference between "still better", "better still", and "better yet"?

I recently got this question on Twitter:

@phrasemix Speaking of still, do still better,better still and better yet have similar meaning?

— Sergio Rodrigues (@serpiro) June 26, 2012

The answer is "no". Here's what each of them means:

Still better

Use this when something was better before, and it continues to be better now. For example, a father might say this when his teenage son challenges him to a basketball game:

I'm not as good as I used to be, but I'm still better than you.

You can also use "still better" in a situation like this:

A: How's your chicken?

B: It's a little dry, actually. It's still better than anything I could have made, though.

In this case, "still better" means "better, even though it has some bad points".

Better still

Use the phrase "better still" when someone makes a good suggestion, but you have an even better suggestion:

You can send them your résumé, or better still, see if you can contact one of the hiring managers directly.

A few notes about "better still":

• It's more common in Britain than in the U.S. Americans use "better yet" more often.
• You use it when 2 ideas are good, but the second idea is better. In American usage, you wouldn't say "better still"  if the first idea was actually bad.

Better yet

"Better yet" is basically the same in meaning as "better still", but it sounds a little more negative.

You can use "better yet" in the same situations as "better still":

Why don't you take a snack with you? Better yet, here's some money; stop and get something to eat on your way.

You can also use it to offer a better suggestion to replace a bad idea. For example:

A: It's not my fault I was late. I got stuck on the train. I'm sorry, OK? Next time I'll try to call to let you know.

B: Better yet, why don't you take responsibility for yourself and leave eariler next time.