Har.
Well, I guess over all then (playing by your rules here, !?) I’d have to go with Cash, although overall I think Sinatra has probably had a greater overall impact on today’s culture.
First, though Sinatra was a hell of interpreter, that’s pretty much all he was: Cash wrote some great, great songs along the way; his strength as an interpreter was not apparent to most of us till later in his career.
Second–personally–I find more depth in Cash’s work; more emotional depth I mean. Sinatra was a technician. Cash was, what, practically a diarist; talk about your heart on a sleeve.
Sinatra was from a tradition of–here’s a songwriter, here’s a singer; never the twain shall meet. He came from the jazz tradition of starting with a familiar tune and then adding some texture, some personality to it; the paradigm was kinda like playwright/actor: division of creative labor.
In that context, Sinatra was a great “actor” of songs–surely one of the greatest (though I too would choose Fitzgerald over all others).
Cash came from outside of any music-establishment tradition. He came from “pop” music–as in popular–as in music of the people. American Country music did not come out of academia, but out of the woods and the bars and the front porches of the margins of society. (So did Jazz, of course, but IMHO Jazz eventually achieved a kind of academic classicism that Cash’s brand of Country seems to have resisted.)
I don’t know why this is important to me, or even how important to me it is. But Cash, it seems to me, invented himself, and kept reinventing himself along the way. Sinatra adopted an existing genre/style, and though he may have come close to perfecting it, he never strayed very far from the path laid out by those who came before him.