I've been having stability issues with Steam Oblivion and was wondering if there are any differences with the GOG version that might be better. I know that there is a different OBSE for the GOG version so there must be some differences. Anyone have any thoughts on this?

I had stability issues with disk and gog Oblivion when I was on windows 7 and it was getting hung up by the virus program causing microstutters.

Since I've been on windows 10 it has run completely smooth. I do not have steam so I cannot tell you if there is any difference.

I'd get to the bottom of existing problems before trying something new - or you're prepared to see the same problems again in the new environment. There is also a difference between standalone Oblivion and Oblivion running under Galaxy (additional system layer, like Steam, and complexity).
I myself have been playing Oblivion from the beginning, CD/DVD versions first, then GOG standalone, in varying PC environments. All non-software problems I had turned out to be selfmade and easily fixable. I have no experience with Galaxy or Steam Oblivion.

Steam sucks.
Only buy from GoG.

Yeah but I bought it before GOG had it.

By stability, I mean frequent crashing, either CTD or hang. I've found Oblivion to be one of the worst offenders ever since my day-one disk installation which was a nightmare that started with not being able to autorun the installer. Maybe it's me but I've had only one stable installation over 6 different computers since then. (not that I've played much in the last 10 years)

That being said, I went ahead and bought the GOG version on sale. I installed the stand-alone version and everything just works (as Todd might say) A few hours in and no problems, we'll see...

It's not you, it's Oblivion.

Oblivion and Morrowind (as well as FO3 & 4) are quite notorious for being unstable and crash prone. Some issues have been fixed by modders, but there are still things that can go quite wrong without it being the user's fault.

Sounds like an Oblivion problem indeed (and you'll very likely encounter this type of problem with your GOG installation, too, sooner or later). Some of the crashes are caused by memory (mis-) management, some by mod incompatibilities (not Oblivion's fault most of the time), some by a number of other reasons.
My personal experience is that the crashes appear rarely during the first hour of a session, and that the probability of crashes rises beyond the first hour (which is why I'm hit almost never - my sessions rarely last longer than an hour). As a consequence, I save often.
By the way, when I'm to mention one particular situation where crashes are most frequent (but still rare) according to my experience, then it's cell changes, for example when going through doors, or when crossing from one region into the next (not always easily recognisable while playing).
If you encounter such crashes during the first hour not only occasionally but frequently and reliably, of if a particular crash is reproduceable, it might be worth considering other explanations.

it crashes just as much on gog Galaxy as it does in Steam... bottom line is the game is old and a lot of the mods are poorly coded so make saves