PSP was released in 2005. PS Vita (aka PSP2 / Next Gen Portable) was released in 2012 in the US.
PSP uses Universal Media Discs (UMDs) while Vita uses specially designed game cards (likely based off SD cards).
Hardware wise...
PSP has:
- 333mHz (single core) CPU
- 166mHz graphics card (with 2MB of vRam)
- 32 (or 64 for PSP-2000 and PSP-3000 models) MB of ram.
- up to 32GB of external storage on Memory Duo cards
- 4.3" (TFT) screen
Compared to the Vita's:
- quad core ARM Cortex CPU (clocked at 2GHz per core).
- quad Core PowerVR GPU (with 128MB vRam)
- 512MB system ram
- up to 32GB of external storage on proprietary Vita-specific memory cards (that are somewhat expensive... )
- 5" organic light emitting diode (OLED) touch screen
These are PSP graphics -
Dissidia Duodecim
These are Vita graphics -
Uncharted - Golden Abyss
The Vita isn't backwards compatible with the PSP on it's own... (You can't use UMDs with the Vita.) But if you bought games from PSN then you can download those games to your Vita and they will work.