What is the difference between using only yolks and whole eggs in a curd?
The yolk also has a fair bit of protein but it is combined with a huge amount of fat as well. Whites, on the other hand, have only protein.

When egg whites cook in a custard they have to be mixed very well, cooked gently, and usually still strained, in order to avoid blobs of scrambled egg messing up the texture of the dish.

Egg yolks are much easier. Some recipes will still call for straining them, I expect to deal with the albumin and chalazae that cling to the outside however, this is not nearly as difficult to work with. Egg yolk have enough protein to gently set a custard and they mix and emulsify into the batter easily.

I will sometimes also include whites when making a recipe that calls for yolks- especially if I have a yolk break while separating and I don't want to throw the egg out. In this case I will mix the batter more thoroughly to ensure there are no undissolved whites. Don't mix the egg whites alone or they will foam and never dissolve. I will also heat the mix more gently and stir more often to prevent scrambled eggs.