I can feel a subtle difference but can't articulate it.

I think we hear more than "feel" a difference, Faldage.

"Unrelenting" sounds more implacable, more indomitable, than "relentless" - just as "uncaring" sounds more unfeeling than "careless".

Anything "un" is totally the opposite of the word to which it is conjoined, whereas "less", by itself, is not an absence of something, only a diminution of it.

Etymologists may agree that "unrelenting" and "relentless" have exactly the same meaning, but they do not 'sound' as though they have exactly the same meaning.

To put it another way, "relentless" sounds as though it is less than completely unrelenting.

Etymologically speaking, the difference is negligible, or non-existent.

Oratorically speaking, the difference is profound.