I've used AWS for a few years at Crate and Barrel and Azure here at my current job. The basic way I describe the differences between the two are that AWS can be a bit more of a low-level experience than Azure. It has a ton of functionality and you can do pretty much anything with either platform, but you will be doing a bit more work out of the box in AWS to get auto-scaling and other services working. I was able to set up Nginx as a reverse proxy and web servers in the backend pool and it worked really well. It was a mix of Linux and Windows. I used virtual machines and auto-scaling groups in AWS whereas in Azure, I did the same thing but with all managed services (App Gateway, Web Apps, Azure SQL). If you are working primarily in .NET, you'll find that the tight integration of all the microsoft products has great support. Linux is supported well too. I'd go with Azure if you are doing development in .NET. Both Azure and AWS seem to have roughly the same availability. I've had outages on both platforms, but they are infrequent.

Now I've got the Monkey Island music stuck in my head.