So, I just got GOTY on Xbox today, and I suddenly wondered- if I play an already existing character that has tons of items from vanilla, and then go and get items from the expansion, and try to play on the vanilla disk again- what happens to those expansion items?
Does it patch my whole xbox? And most importantly, what glitches would it fix?

Does it work like shivering isles (install once, don't need the disk) or do you need to play on the goty disk to use the goty content?

I don't play on a console, but on PC you always use the Morrowind disc to start the game. I assume it works the same way on XBox.

The Morrowind expansions work just like the Oblivion expansions - all your existing items will still be there after you install the expansions.

A GOTY Save file would not be able to work on the old vanilla disc.

And why would you want to do that anyways? GOTY updates the game and fixes a bunch of stuff, too.
Use the old version only if you have no access to the GOTY version.

So this means that if I loaded my pre-existing character on goty version that I could never use him on the vanilla again?

Correct.