What are the main differences when using
I think the Mobile Service is mainly focus on quick mobile service development especially for those data oriented. If you want to create a mobile application quickly, with CRUD against your data on cloud, with push notification out-of-box, mobile service might be a good choice.
But if you need complex server side logic, Azure Web Site could be better. You can create your own API with your own logic by using ASP.NET MVC WebAPI, Node.js, etc.