The role of fat (butter, oil) in caramel-required recipes?
Well, I'd say it has mostly to do with texture, but also with taste.

To my knowledge there are three basic methods to making caramel: 


dry:  without anything but sugar. In my opinion the most difficult method, as it tends to burn easily.
wet: with water added to make sugar syrup, which is then turned into caramel. The water evaporates during the process. It takes a bit longer, but in my opinion definitely easier to handle than the dry method. Also, as i recall, has a slightly different texture than the dry method caramel.
fatty: with fat, obviously. Fat doesn't evaporate and can get very hot, even though, not as hot as sugar alone. For me the easiest method. However, it should under no circumstances come in contact with water, or it get's very, very dangerous. 


So in the first two cases you end up with mostly caramelized sugar and maybe water (depends on whether you let the water evaporate completely or if you try to end up with caramel sauce). With the third method, however, you end up with a caramel-fat mixture, which has of course different properties. For example, if it has enough fat, it turns soft again if it gets warm. 

I think, for most of my applications, the fatty version gives a smoother, softer texture, which could indeed aid with the coating of nuts for example. This is probably, because most fats are soft or liquid when room temp. I think, it also affects taste, because fat is a natural flavor enhancer, which brings out the caramel flavor even more. Just like the salt in salted caramel. (On a side note, i think it also looks shinier, but that could just be my imagination. ;) )

As always with caramel: Be extra careful whichever method you're using. It is soooo fricking hot and you can burn yourself very, very easily. Stay safe! 

Edit: for some extra science take a look at this paper I just found. It examines the effect of fat on caramel texture attributes. Might be interesting!