I would say there are two main differences and then some smaller more subjective ones. The main two:
Syntax - Elixir has a more familiar looking Ruby derived syntax, plus some little extras like the pipe operator. In practice, once you've used either Elixir or Erlang for a few hours the syntax difference isn't that important, you get used to it quickly. That first few hours is important though - if someone doesn't get through them they're not going to learn the language. Elixir makes that a bit easier, gives you one less thing to think about during early learning and helps adoption.
There are some other niceties like the pipe operator and for comprehensions that can help organise code, but you could live without them.
Macros - Erlang has no capability for metaprogramming. Macros do need to be used sparingly and carefully as they can obscure what's going on (Erlang is very literal and that's a good thing). When used well they can be a great help at avoiding boilerplate. Elixir itself uses them for things like GenServers and supervisors. This does things like providing default child specs and allowing them to be overriden in the GenServer. Phoenix makes heavier use of them, mostly seamlessly (and it's a great benefit). The router is the only place where they're slightly confusing as the magically macros generate some extra helper functions. The mix tasks (mix phx.routes) show these generated function names, but it's not the same as having a definition in a file.
Other hand wavier things include better tooling (although a lot of the improvements are now present in Erlang tooling as well) and better documentation.
Syntax - Elixir has a more familiar looking Ruby derived syntax, plus some little extras like the pipe operator. In practice, once you've used either Elixir or Erlang for a few hours the syntax difference isn't that important, you get used to it quickly. That first few hours is important though - if someone doesn't get through them they're not going to learn the language. Elixir makes that a bit easier, gives you one less thing to think about during early learning and helps adoption. There are some other niceties like the pipe operator and for comprehensions that can help organise code, but you could live without them.
Macros - Erlang has no capability for metaprogramming. Macros do need to be used sparingly and carefully as they can obscure what's going on (Erlang is very literal and that's a good thing). When used well they can be a great help at avoiding boilerplate. Elixir itself uses them for things like GenServers and supervisors. This does things like providing default child specs and allowing them to be overriden in the GenServer. Phoenix makes heavier use of them, mostly seamlessly (and it's a great benefit). The router is the only place where they're slightly confusing as the magically macros generate some extra helper functions. The mix tasks (mix phx.routes) show these generated function names, but it's not the same as having a definition in a file.
Other hand wavier things include better tooling (although a lot of the improvements are now present in Erlang tooling as well) and better documentation.