It is not exactly the same, however the PS1 emulator in Vita is based on the PSP one (they updated the emulator to improve compatibility and stuff, and it seems that the Vita version has access to a greater CPU speed).
Why? Because the CPU of PSP (MIPS) uses the same architecture of PS1, while Vita uses a different architecture (ARM), so emulating PS1 on PSP is more efficient than emulating PS1 on Vita. And Vita already had to include the original PSP processor anyway to run PSP stuff, so why to write a completely emulator from scratch if you can use the PSP one?