I would argue that Heroku is still the fastest way to get a web app deployed for the first time. One command to push and one click to setup a solid database.

As your systems get more complex, you can still peel off services to AWS or the like, but until then, Heroku combines the following tasks and AWS services into a dead simple process: configuring instances, Scaling Groups, RDS, Cloudwatch (some metrics, consolidated logging) and ELB.