Well, for one thing, if you meant a ds flash cart, you can't play GBA um...backups with just that, you have to have another cart for the gba slot. Unless you're talking about a GBA cart that also has support for DS games, which, obviously, would work for both. For the PSP, if it's hacked, you can easily put backups on a memory stick and play them (resulting in better loading times and battery life btw).
As for PSP vs. DS, if you intend to hack the PSP, then it can play GBA games (emulation is nearly perfect, you can thank exophase for it), PSX games (hit and miss, most games work, but some don't...but the ones that run run at fullspeed), (S)NES games, etc. and has a huge homebrew community. The DS can emulate NES and SNES and a few others fairly well, has some homebrew, but not as much as the PSP by any means. If you don't care about that stuff, and/or don't want to hack the PSP, I suggest you just look at the library of the DS and PSP and see which has games you prefer. Gamefaqs or ign or something might be good for that.
Personally, I have both a DS and a PSP and I don't think I could part with either of them. But, I'm a handhelds person, I'm not that crazy about consoles (I never have time to play games at home during school).
If you go with the PSP and plan to hack it, I would say buy the PSP-1000 off of ebay or somewhere. It'll be cheaper, and IMO it's better for its IR port (which can be used as a universal remote control with irshell), better build quality and higher compatibility with older homebrew. And if you go with a DS, I would say get a Lite. It has better battery life than the DSi, and the DSi's future in terms of flashcarts is questionable, nintendo could release an update to make the current ones incompatible, and then you'd have to buy another. Unless you're interested in DSiWare (I'm not in the slightest).