It doesn't really matter, in essence, they're different games. They tell the same story, I suppose, but there are some key differences between in it between the two versions.
The browser version offers more gameplay variety - most is platforming, of course, but there's also motorcycle riding, swimming, a fighting game-like stage, a puzzle game-like one... And it's neat that you can "save" your game if you make an account.
In the HD version, there are just platforming and a few swimming stages. But it's more of an action game, you can do combos (after you buy them from a store), block attacks and stuff. Boss battles are great fun, as is playing couch co-op. Stages sometimes offer different paths you can take to complete them.
I suppose you could play the browser version first to see if you're into the art and the world and everything. It's free, after all. And there's a demo for the HD one available, so you can try that one before you buy, too.
--
Blitzball fan? Try Captain Tsubasa II (in English) for NES!
Game reviews/articles by me: http://betweenlifeandgames.com