The DS came first. The DS Lite presents a more enjoyable gaming experience, as Nintendo incorporated a lot of feedback from DS users, and came out with a better device.
Key differences include:
1. The DS Lite is smaller (surprising, huh?). And smaller really is better. It's more portable, easier on the hands. It's 21% lighter than the original.
2. Better screen. The screen can get much brighter than the original DS, and this makes a difference when playing in sunny conditions.
3. Longer battery life (though I couldn't find out by how much, it seems to be generally accepted to be true).
4. Lots of minor tweaks (moved buttons, different size stylus) to make it more ergonomic.
It's fully compatible with both the original DS, and the Gameboy Advance (as is the original DS), so the library of available games is large.
I've got one and I love it :)