Well, PJ64 1.5.1 is the best version. It is the most advanced, more compatible, etc. One thing it doesn't have, however, is the netplay function. Netplay is the ability to play other people online. PJ64k 0.13 is PJ64 1.4 with netplay (following all this?). Jabo, the developer, created after many releases PJ64 1.5.1. He kept the source code to himself, and just released the program. However, upon releaseal of 1.5 (and the subsequent service pak (the extra .1 at the end)), he decided to release the source code for PJ64 1.4. hotquik, a developer, got this source code and modified it to include netplay. (still following?)
So:
PJ64 1.5 (with SP1, effectively 1.5.1) - The most advanced PJ64
PJ64k 0.13 (using PJ64 1.4, effectively PJ64 1.4 netplay version 0.13)- Not as advanced as 1.5.1, but has netplay.
I'd also like to note that PJ64k is being worked on, while PJ64 is not.
As far as which version you should be using, if you want better quality gameplay, better graphics, and more compatibility to roms (games, don't ask for the roms/games tho, it's illegal), you want 1.5.1. If you want to fight people online, then get 0.13. Needless to say, everyone has both, one to play when they want to play by themselves, and one to play others with. I might like to add, check out my Project64k Support Webpage (signature), it answers any questions you may have and ever have. If it doesn't ask me, and I'll add it to the site. Any help on 1.5.1, and you post in this forum.
Lastly, Have fun! =D
