The X51v has a higher resolution screen, faster CPU, more RAM, and 16MB of dedicated video RAM on its Intel 2700g GPU. Specs alone, you'd think it was a far better choice than the PSP for gaming.
How you plan to use the unit determines which one you should get. The PSP is perfect for playing games with its plus shaped directional pad and game button placement, plus it has tons of games made just for it. The PSP can browse the web in wifi areas and be used for chat etc.
The X51v is not made for gaming, it's made to do everything. You can game on it but the D-pad isn't the best for control. I prefer stylus controlled games on the Axim. I installed Quake and Doom on my X5 and they played OK but the D-pad isn't meant for them so it was hard to control. I installed Toy Golf and the racing game made to be played on the X51v's graphics card and while they looked pretty and were stylus oriented, it wasn't controlled by the plus shaped D-pad and well placed game buttons the PSP has.
The Axim is a mini computer meant to do many of the things you can do on your desktop computer. The PSP is a very good gaming handheld with limited internet functionality. Which do you want? (there is no wrong answer, this is how you decide)
BTW,
Aximsite has a great writeup of games made for the 2700g chipset on the X51v.