I think Emrys is a trendier and more artsy cool name - it’s more on keeping with names like Felix, Tolliver, Auden, Leander, Alistair, Jasper, Osian and has a more enchanting, fantasy novel feel.
Rhys (and I definitely prefer this spelling for guys) is a confident, effortlessly cool, boyish name - in keeping with names like Liam, Tristan, Finn, Tobias/Toby, Rory, Wyatt, etc.
Both swish names! Emrys sounds more handsome and buttoned up, Rhys cheekier and a little more cocksure. I like the sound of Rhys better but the meaning & debonair style of Emrys are real pluses!