I've used both over the years.

Got a bunch of hide pieces from a taxidermist a decade ago. There was this piece of really light colored deer, still brown, not belly hair, that I started using for EHCs just because I could see it easily. Flares just enough, floats well (but I waterproof all my dry flies, so I can get almost anything to float.) Haven't used anything else since. I have enough in this one piece to tie all I'll ever need.

The thing is, it seems to not matter to the fish whether we use deer or elk.

As long as it's what you want or need to use, and it works, it just doesn't matter.