I have recently migrated an application from Heroku to Amazon EB and am missing Heroku already :)
Application wise, everything is the same. The major differences regard pricing, performance and ease of operation.
1. Price - Amazon will beat Heroku for nearly anything above two servers.
2. Performance - again, Amazon gives you the flexibility to deploy your app on a RAM monster, speeding things up significantly
3. Operation - that's the tricky part where Heroku wins big time.
Operations wise, Heroku is extremely easy to configure and maintain, while in Amazon you need to dig deep into the Elastic Beanstalk deploy scripts in order to customize them for your application. Deploying isn't a breeze as well, as EB CLI isn't as easy to use as Heroku and forces you to keep thinking about the underlying infrastructure (region, vpc, load balancing, security, etc).
I'd stick with Heroku if you're happy with price/performance and move to Amazon only if you really want these parameters notched up.