Standard (BBC newsreader) British (when in the UK I pass for a posh native!) definitely distinguishes them. It's Muriel with a You, or Mew like a cat, and Mariel either Mare- like the female horse or Ma-ree-el, with a very short final syllable and the first vowel like the one in At and Cat. In fact, when I say it, it's Ariel, like Sharon, with an M.
A lot of people seem to dislike the Muriel version, so it could well be that Mariel isn't so much a confusion of sounds as a conscious avoidance. I think Mariel sounds much better, though Muriel belonged to two beloved relatives so I've got no bad associations.