I own Skyrim and the Oblivion GOTY on both console and PC.
I recently got back into Skyrim and today I bought Oblivion GOTY.
The first thing that came to my mind when playing Oblivion was, "this is a fricking eyesore" (especially character creation, youch!), and that turned into me willing myself to keep the disk in and continue on. Soon enough the graphics grew on me and became bearable, so that was at least out of the way.
The first difference is the class and leveling system, which some like and some were more than happy to get rid of. For me, I personally like it along with Skyrim's system. I feel like Oblivion's system is superior in the whole RPG aspect while Skyrim's is good for those who like their characters to be a jack of all trades or a moldless base for their own... whatever type of character. Leveling is more "realistic" from what I can see, as leveling the major skills of your character's class will get your overall level up quicker.
Other noteable differences would include the guilds (which I have not delved too deep into yet) and quests (obviously).
Overall, while the presentation is different, Oblivion/Skyrim have similar mechanics and lore (just different time periods)... Oblivion, in my opinion, holds a bright candle to Skyrim. I think that if you love Skyrim and you picked up Oblivion and gave it a valid chance, you would enjoy it very much.
If you have any specific questions please hmu over PM!
EDIT: I also must add that the interface is absolutely atrocious. On one hand, I'd recommend Oblivion for PC bc of various interface mods and on another I'd recommend console for controller support.