Congratulations of finding those.
I had a DS (friend gave it to me as a gift a few years ago when he upgraded and it was my first game system) and have a DS lite which I got as a replacement so I can speak on both.
Coming from the perspective of someone with shaky hands, the DS is built like a tank. So if you're like me and have a problem dropping things, the DS will be great.
Both play GBA games like everyone has said, so there's no real difference there, just the feel of them.
For games, I got leaf green on ebay (first game I ever owned), and I have Diamond, Black, and a few other games like Legendary Starfy which is a good game for Kirby fans. I had Kirby and may try to get my hands on it again in the future when it's possible.