I'm a newb but the differences I've seen so far have to do with diver confidence and buddy behavior. I think if you're comfortable with your own skills in warm water you should be in cold water as well, but that is often not the case. I find, for me, I have to consciously think more about controlling my breathing in cold water (especially diving wet) and slow down in general. In warm water it's just easier to be physically comfortable and thus mentally comfortable.
Buddy diving in clear water tends to make people complacent about buddy distances. Too complacent, in my opinion. Diving in poor viz you really need to be aware of that distance and keep it reasonable.
If you're a conscientious diver trying to be a good buddy you can certainly do well transitioning from warm/clear to cold/low viz but I think it's generally easier to go the other direction. It's certainly possible to transition either way, though, especially if you plan for it in advance.
Buddy diving in clear water tends to make people complacent about buddy distances. Too complacent, in my opinion. Diving in poor viz you really need to be aware of that distance and keep it reasonable.
If you're a conscientious diver trying to be a good buddy you can certainly do well transitioning from warm/clear to cold/low viz but I think it's generally easier to go the other direction. It's certainly possible to transition either way, though, especially if you plan for it in advance.