I have to agree with Jake Peters. The best answer is AWS if you're comfortable with running LAMP stack servers, relational database servers, elastic load balancers, auto-scaling, etc., which requires a ton of knowledge, especially the way AWS is setup.

For simplicity, Heroku is unquestionably the best option if you'd prefer to spend your time working on marketing your app than learning the ins and outs of how AWS is setup because you'll be forced to learn things like:

What's an EC2 Instance?
What's an RDS Instance?
What are Elastic IPs?
What's IAM?
What's an ELB?
What's an AMI?
What's a bucket and more importantly, a bucket policy?
Am I using the right kernel?

I can go on and on. AWS provides the most flexible, scalable and robust portfolio of services and options, all at your control, but there's a learning curve, for sure.

Once you get comfortable with AWS, you will never want anything else. This is why Netflix, Pinterest, Instagram and a host of others use AWS.

Heroku will help you cut the learning curve and get your app up and running so you can spend time on the most important part of app development: marketing.