Both of us would be me. Then, as our experiences diverged from the point of duplication, we'd become different people (See: Thomas, the duplicate of William Riker in Star Trek. The only reason Thomas and not Will is considered the copy is because of audience perspective, but empathizing with each of them makes one see how both are Will Riker at the start of the episode). This all of course, assumes we don't discover something like the popular conception of souls during the early trials. But I don't believe there's anything about a "soul" that can't be tied to the sum of one's lived experience, which would be copied too.

I would consider a clone to be more expansive of who it could include besides copies of myself as I am now - it would also be someone grown from the literal same embryo as me who'd lived a completely different life with even a different name.