No matter who did it first, Elvis sang it better.
This applies not just to songs like "Blue Suede Shoes" and "I Just Can't Help Believin'" but even to "Bridge Over Troubled Water," and as anyone who reads this site regularly (which should be everyone, at this point) knows, I adore Art Garfunkel. But he's not Elvis Presley.
The exception: I like Elvis' version of "Always on My Mind," but it does suffer from a certain lack of gravitas, which this, one of the saddest songs in the modern canon, deserves. And the background singers, which I assume are some incarnation of the Jordanaires but sound here like the Oak Ridge Boys on a coffee break, are a joke.
The definitive version of "Always on My Mind" is Willie Nelson's, from 1982, which went to Number Five on the pop charts and Number One on the country charts. Willie is rueful, ruminative, at the peak of his interpretive powers, and it doesn't hurt that he was already 48 when he cut it, at a time when one should be looking back on life and regretting treating your one and only as if she were second best.