The most significant new feature is savestate support. You can now save your progress at any point, via the Pause Menu (accessed through hitting the Select button). Savestates are written out to the memory stick, and consume around a megabyte per slot. You can load up a savestate at any time from the Pause Menu, or via the front end (hit the right shoulder button to swap from the rom list to the savestate list.)
Other than savestates, the most significant change in R13 is a number of optimisations to the dynarec core which should give a 10-20% speedup depending on the title being played. I've tested these optimisations as much as I can, but if you find that roms which worked with R12 are now broken, try disabling 'dynamic recompilation' and/or 'dynarec stack optimisation' from the rom's preferences screen.