The difference between a developer advocate and a developer evangelist is inbound vs outbound.
Developer advocates are focused on helping address developers' problems. Developer evangelists are focused on increasing product use and adoption.
That said, the responsibility of these positions mostly overlap. In my personal experience as a developer evangelist at Marketo, my day-to-day work has encompassed the function of both titles. My goal is to create a great developer experience [0] in order to drive platform adoption [1].
Anecdotally, I have noticed a shift towards the developer advocate title. At Facebook, Dropbox, and Twitter, the role is called this.
[0] http://murtza.org/creating-a-great-developer-experience/