Do these mean the same? If not, what is the difference in translation and grammar? If you can, please give me some examples. Thank you!
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To me they mean the same, but I'd say "still better", not "better still".
I would say "still better" too because it has the right order in my language.

Question for native speakers: Does it have to be in the right order?
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The Economist often uses "Better still". It is definitely not incorrect. Here are some examples:

"Better still, Babylon is capable of translating completely novel sentences."
"Since then some figures have got better still."
"Better still, Mr Blair was a true believer, exuding conviction."
I never said it was incorrect, I was just stating my opinion Emotion: smile
"Better still" and "Better yet" are both used idiomatically to mean "Even better than what was just discussed." There will be a lot of chocolate at my birthday party, and better yet, a sundae bar.

"Still better" would be used as follows: I'm not very pleased with this old dress, but it's still better than the ones I'm seeing in the stores now. Or: I know that Dean Martin songs are "oldies," but he is still better than that stuff you hear on the radio these days.
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So you wouldn't say: 'I know you like oldies... Here's a Dean Martin, and, still better, an Edith Piaf.from the 30ies" ?
That doesn't sound natural. But that's just my American ears. Others may disagree.
Oh no, GG... It's just my non-native ears Emotion: smile
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