I compared Authorea and ShareLaTeX early on and have been subscribing to ShareLaTex for several years. The biggest selling point at the time was the option to setup a ShareLaTeX server at any point when I got around to it. That, and the track changes in ShareLaTeX when working with my team is its biggest selling point for me at the moment. However, my institution recently signed up with Authorea so it may be that they have been able to catch up in many respects to convince the decision makers here (or else, on the pricing).