Here are, as I see it, the main differences:
He'd be more vulnerable than the ordinary high-end brick against any purely magical effect. For example, a magical pain spell is something Thor would shrug off, but it would drop Supes like a bad habit.
Magic weapons will cut through his durability much better then their equivalent counterparts.
Now, you have to remember that the whole "vulnerable to magic" thing came from pre-crisis, and back then, he fought normal dudes with magic knives and their knives cut his hair. Spells being cast around him and missing caused him general physical weakness and outright sickness at times.
DC wrote a dude who was very powerful. To keep new stories coming, he kept getting stronger and stronger and stronger, until he wound up at "effortlessly tows strings of planets around on a chain" level. Initially, he had kryptonite as a (completely stupid) weakness, and being around it more or less dropped him instantly. He didn't fight through kryptonite back in the day, it just dropped him. Of course, they couldn't just have every enemy in DC carry a rock of K around with them at all times, so they created another way to put him on the ropes.
Post-crisis, they notably weakened him, so they backed somewhat off the "Kryptonite drops him instantly and magic is almost as bad" levels, and had him start fighting through those things better than before.
History of DC. :-)