Technically, the latest version IS pj64k v0.13 core 1.4 (created by hotquik) which is the exact same thing as pj64kVE, a port of pj64 1.4 (created by Zilmar and Jabo) to increase compatibility with Vista when it first came out (also works with 7). However, it still uses the 1.4 core and version 0.13. Some people may not have compatibility problems on Vista or 7 with the regular pj64k, so they could use that instead. kVE was just offered as a slightly better alternative to help cope with the crashing (not perfected, though).
pj64k7E
The pj64k7E is, to my knowledge, the same thing as pj64kVE, and the guy just picked it up and started messing with the code in a hex editor to improve a few things, but it still remains as v0.13, core 1.4. So, technically nothing has changed and it should be no different from kVE. You can even look at the changelog, and observant eyes will notice that he really didn't do anything to warrant the new name and touting it as this great new alternative to kVE.
The only thing he fixed (supposedly; unverified), was that random error some people get when loading pj64k that says the plugins can't be located. Bear in mind that this is easily correctable by going to settings and choosing the damn plugins yourself. Takes all of 10 seconds. He might've also changed the way kaillera loads plugins, and will allow you to load different graphics and sound plugins without renaming DLLs. Again, this is something you can easily do yourself (rename any graphics plugin to Jabo_Direct3D7.dll or any sound plugin to AudioHLE.dll and you'll load those plugins when starting netplay). It also looks like he hasn't even included Owna's updated client (ew).
pj64k v0.34
pj64k v0.34 is smasherx's slightly more modest project, in that he didn't opt to rename it completely, and continued with the same version "edition" scheme (the proper lingo eludes me right now), improving from 0.13 to 0.34. Again, the things he improved are basically the same as pj64k7E, so it only improves convenience slightly. He also mucked about with the netplay client, renaming it and calling it differently, however still retaining Owna's client. Not of much concern.
Conclusions and Comparisons
To summarize, all of these seemingly different versions of pj64k are exactly the same. Regardless of what emulator you're using, when you start a game in netplay, the kaillera client lists your emulator as "Project 64k 0.13 (01 Aug 2003)". Right here you can clearly see they're all using the same 0.13 version the original "latest" version of pj64k uses. You wouldn't be able to play with somebody who used a different emulator. Nobody changed anything major, they just made small changes akin to grammar fixes. It is, in essence, a prettied up pj64k 0.13 core 1.4. There are no huge developmental breakthroughs that will decrease your lag or desyncs. They are exactly the same. Any changes you noticed in desyncs were purely coincidental and fall under some psychology term for you wanting to believe there is a difference and seeing it. There is not. Don't let anybody fool you that there is.
32-bit vs 64-bit
All of these emulator versions were designed for x86 systems (again, the version these are all based off is from mid-2003 when x86 systems were largely prevalent). So there will be compatibility problems with x64 machines, but I don't think there'd be much difference aside from a few weird memory errors or crashing. Certainly nothing game-changing like increased lag or desyncs. It doesn't matter if you're on on 64-bit 7 playing somebody on 32-bit XP; the experience will be the same as two people playing on identical machines.
Actually, just looked it up. I can confirm pj64k was initially designed for 32-bit systems, so everything discussed here is 32-bit.
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The only reason we continue to use pj64k and pj64kVE is because they are tried and true and as "official" as you can get with unofficial software. Not sure about this pj64k7E Gaudy character, but smasherx is known to be "malicious" in how he goes about things, so as a standard rule of thumb, we ignore anything he outputs, lest it contain a virus or whatnot. You're welcome to try and use these newer versions, but remember it's at your own risk and we will not help or provide support for anything other than pj64k(VE).
Possibility of Change
The only reason we'd change is if hotquik continued developing pj64k. I'm not sure why they stopped where they did at core 1.4 when 1.5 and 1.6 were released shortly after, but it may have been due to source-code issues with Zilmar and Jabo, or they simply didn't have the time to continue such a project. Who knows, but the only updates we would accept as an improvement would be from the hotquik team, as only they have the source for pj64k 0.13 and only they can improve upon it.
pj64 1.7 incorporates a reliable netplay, so once that is released to the public, I imagine that will be the netplay standard.
Now, there is also Mupen! The latest version is Mupen64++ Beta 0.1.3.12, and this supports netplay and has been empirically proven to reduce desyncs. However, Mupen requires a fair amount more computing power than pj64, and is a little more advanced, so it hasn't quite garnered the attention it deserves. Everybody would have to switch over, and it would be difficult to encourage, people would rebel, be pissy, etc. and it's generally not worth the hassle. However, if you and your friend are looking for a smoother ride and have relatively powerful (by 2005 standards) PCs , try using Mupen.
Phew... that was a lot, but I hope anybody who sees this will realize what's really going on and not let the wool be pulled over their eyes!
TL;DR
Despite what you or others may have you believe, any version of pj64k you find is the same. The only differences are minor convenience fixes, but nothing that will alter your online experience. We all recommend pj64k or pj64kVE found in the index, as they are the most reliable and supported by our ****ty tech support.