Here's a quick recap:
DS(lite) advantages:
1) cheaper
2) Right now it has access to many,many more games (I don't know exactly how many Game Boy Advanced games there are, but it's probably in the 1000's) through the GBA port.
3) Because of the access to these games, you can find really cheap games to play with it (For example,
Amazon has hundreds of GBA games for great prices)
DSi advantages:
1) faster / more powerful processor (this is something that will be used when they start making DSi specific games)
2) larger screens (3.25 inches vs. 3 inches for the DS(lite))
3) Internet connection (requires wireless router)
4) 2 digital cameras
5) uses SD cards (Up to 32GB. Can transfer photos to and from the Wii or a PC)
6) Can be used to play music (it doesn't use MP3's, but rather AAC files which is actually a better system (although not as popular as MP3's))
7) You can download programs and games through the DSiWare program. kind of like the Wiiware for the Wii. So far not much has come out, but there are high hopes of some great programs being available.
8) Future co-op use with some Wii games.
9) Future access to Nintendo handheld games via DSi Virtual Console. They are talking about making all old Game Boy and Game Boy advanced games eventually downloadable (thus eliminating advantage numbers 2 and 3 for the DS)
10) improved speakers
So basically, right now there isn't a huge advantage of a DSi over the DS, but in the not to distant future, I think you will see many really good advantages.