I just noticed that they seemed to have changed the cover for the UK blu-ray of The Thing. Before, it was the movie poster art. Now, it's all white. I'm assuming it's the same disc. But I'd really like to get the version with the original poster art cover.

ebay or used copies would be one way.

Or get one from another country. The US/Canadian disc, Japan, Germany, Spain, etc has the original art.

It has to be the UK version. It's the only version that has everything it's supposed to have.

The European and Japanese discs are exactly the same disc as the UK version.

How can I get one of those? I mean, which site? Would the extras work in an american player?

They are all the same disc physically and are region free as usual with Universal discs. Check Amazon.

Amazon DE (Das Ding aus einer anderen Welt )
Amazon ES (La Cosa)
Amazon FR (The Thing)
Amazon IT (La Cosa non fidarti di nessuno)
Yesasia HK (The Thing)
Amazon JP (The Thing 遊星からの物体X)

What do other versions lack?

The US version has the "Terror Takes Shape" 80 minute documentary as a picture-in-picture during the main feature in the corner of the screen.
International releases have both the option of watching it in picture-in-picture, or by itself in full frame, which is more preferable.

Also, the international versions have lots of galleries, outtakes, screentests, and the trailer, which is not on the US disc.

The US version doesn't have the entire Terror Takes Shape documentary. That's why I'm looking to get the UK version.

Get one of the other international ones listed or get the UK version and print your own cover. It's not that hard.

EveryTHING.