As far as I remember, Elm has a runtime that at the moment requires the browser api. Purescript just compiles to javascript. So you don’t have server-side Elm, but you do have server-side Purescript. Which probably makes it more like Reason than like Elm.
The main difference is, Elm is like a drastically simplified version of Haskell, while Purescript is essentially Haskell with benefits (row polymorphism).