This page outlines the differences between the Brilliant Diamond and Shining Pearl versions.
All mainline Pokémon games are released in two (or more) versions. Both versions are essentially the same game, so you only need to buy one (unless you’ve got siblings etc. who also want to play, for instance).
The main difference between the two versions is that some Pokémon only appear in one particular version or are more likely to appear there. This is meant to encourage trading between players, because you can’t get all the Pokémon with just one game.
Exclusive to Brilliant Diamond
Exclusive to Shining Pearl
Which Version to Pick?
With the advent of online trading, it’s fairly easy to obtain any version exclusives you want, except for the Legendary Pokémon. Because you only get one of each and they’re extremely powerful.
Therefore, if you’re undecided about which version of the game to buy, get the one with the Legendary Pokémon that you want. Unless you can’t trade online; in which case, follow your heart.