I have asked around and nobody seems to know!
I have a phobia of moths and just felt like I needed to know! Thanks
Moths' furriness comes not from fur or hair, but from scales. Its wings are covered with small scales, and its body is covered with longer versions of the same thing. They may be different in structure to hair, but on the body they serve similar purposes - to act as an insulator and to form part of the overall pattern of the creature (used for camouflage or deterrence amongst other things). Some scales have special purposes, eg. androconial scales on males, which are concerned with pheromones (they can occur on wings or body).
It is often said that, with moths, the insulation factor is important because, in order to fly, their muscles need to be warm; they need to retain the heat generated internally.
I wish it were actually that simple...
If it was that simple, species that fly in the colder months would have the 'furriest' bodies and summer-flying ones the least 'furry'. One would also expect that the smaller moths would be 'furrier' than bigger ones, because of heat loss factors (ratio of volume to surface area); that is not the case; perhaps the assumed model is too simplistic.
Butterflies are partly heliothermic - they get some of their warmth from the sun. That may explain why they tend to be less 'furry' than night-flying moths.
One would also have to explain how other night-flying insects, such as ichneumons and mosquitoes, manage just fine without all those scales. Yes, different metabolisms come into play, but nonetheless it needs consideration.
So, there are other factors at play, and I don't think we know the proper story yet.
There is a good deal of relevant information online, eg. this about thermoregulation, but I am not aware of a good synthesis of it that gives a satisfactory answer to your simple and good question.
If somebody here knows otherwise, please enlighten me!