You probably know everything about Pokémon Diamond and Pearl at this point so why don't we just go from how Platinum stands above them? First of all, Platinum introduces new forms of Pokémon like Rotom, Shaymin and Giratina. These new forms aren't just for comsmetic purposes, their stats actually change so you can use them for different strategies.
The stories of Diamond and Pearl blend into 1 and go much further in Platinum, allowing you to encounter and capture both Dialga and Palkia, enter the Distortion World, and even find legendaries like Moltres or Regirock rather than migrate them from GBA games.
With WiFi, you'll be able to play mini-games with friends or record battles and play them back whenever you want, or provide a code to others to use to watch your battle on their DS.
There is a completely new GTS server for Platinum players only which I would highly recommend against using since many WiFi players these days use cheating devices to trade fakes. Through WiFi you can also download events like the current Darkrai event (August 2009 - check pokemon.com for details) as opposed to the former system where players had to go to GameStop or Toys R' Us to get a rare legend. So you can download the ticket right from home and access Newmoon Island and capture Darkrai yourself.
There are special move tutors to teach your Pokémon all kinds of new moves as well such as a Blastoise with Signal Beam.
There's a lot more to do in Platinum so its definitely a wise investment as opposed to buying Diamond or Pearl which is only a few dollars cheaper.
Update On Aug 13, 2009: Battle Frontier is introduced and expanded upno to include different kinds of battling events like Battle Castle and Battle Roulette.
After completing the game the Elite 4 actually becomes stronger.
Almost every town and structure has been upgraded as well such as Amity Square having warp panels and really just having an entirely different look from the D/P version.