Unrelenting is constant. Relentless may pause, but will never give up.

Interesting, plausible theory, Jackie, but not supported by any dictionary I have consulted.

For instance, the American Heritage Dictionary [see below] does not recognize any difference between "unrelenting" and "relentless", and it contradicts your suggestion of a "pause" in persistence.

In fact, it defines "relentless" as persistence which is "unremitting".

The kind of persistence you are describing is intermittent.

re·lent·less ( P ) Pronunciation Key (r-lntls)
adj.

Unyielding in severity or strictness; unrelenting: relentless persecution.

Steady and persistent; unremitting: the relentless beat of the drums.

P.S. I agree, however, that there should be a word which describes relentless intermittent persistence ... like a viral infection which succumbs to a new vaccine but mutates and comes roaring back years later, as virulent as ever.