Specifically, what is the difference between caramel sauce and caramel syrup? Whenever I see someone make a caramel mocha latte or any caramel coffee, they either flavor it with the sauce or with the syrup. What the difference? And is it better to use one over the other when making an iced coffee vs a hot coffee?
Caramel syrup, as the name says, it is a syrup, so it is composed mainly of sugar and water, then cooked to concentrate the water and caramelize the sugar.
Caramel sauce on the other hand, usually contains heavy cream or other sort of fat-heavy component (some people add butter, or vegetable cream) to make it similar in consistency to a ganache.
The application depends on the intent.
If you mix caramel syrup in iced coffee for flavor, it will not mix well and remain as a layer in the bottom. Caramel sauce is easier to mix and dissolve, and it also has a richer flavor because of the dairy / fat.
If you want to use it to decorate your coffee (hot or iced) on top of the whipped cream, I'd recommend syrup because it holds well in lower temperatures and it is better to make drawings.