In the most common case they are completely indistinguishable — and in recent years they’ve come to be used with comparable frequency, as I show in the graph below.
That said, there can be differences between them in certain “rescue readings” given enough set up, but these differences are ones that Google N-gram is useless for distinguishing because it takes more context.
Mostly the differences are because yet and still do not themselves work quite the same way, and so even when you mix better into the equation, there can still sometimes be a few minor but lingering differences remaining.
For one thing, you can swap the order to change better still into still better much more easily than you can change better yet into yet better. The last of these does not work:
Better still, do it yourself.
Better yet, do it yourself.
Still better, do it yourself.
*Yet better, do it yourself.
Notice the difference in magnitudes here in this revised Google N-gram:
Another way to look at that is like this, showing ratios of 18 ∶ 19 ∶ 1 ∶ 12, respectively:
- 36.2% for better yet
- 37.8% for better still
- 01.7% for yet better
- 24.2% for still better
You can say this easily enough:
Still better would be making it attractive to people to be less reliant on cars and to use less carbon-intense forms of transportation, especially for short journeys.
But you cannot do that with yet, at least not so naturally.
For another thing, yet is a negative-polarity item, as John Lawler explains here and here. So these are ok:
Are you better yet?
I’m not better yet.
But these are not:
Are you better *still?
I’m not better *still.
However, you can put the still first when you have some follow-on bits:
Are you still better than he is?
Best Answer
I think there's a subtle difference, although we don't give it much thought when speaking.
I would use "yet" if you're adding an element of surprise.
He likes dogs, yet he is allergic.
I would use "but" otherwise.
He likes dogs, but he dislikes cats.