RDS is a managed solution. Which means, AWS staff will take care of:
- Patches
- Backups
- Maintenance
- Making sure it's alive
Hosting your database in a second EC2 instance means that:
- You have to manage everything of the above
Using a LAMP stack and co-hosting Apache and MySQL is the cheapest, but:
- You have to manage everything of the above
- You're probably hosting a database on an instance exposed to the internet
That said, if you're planning to host a production website / service that's more than a personal website / blog / experiment you'll probably need to host webserver and database in different instances. Picking RDS is less of a headache.
For anything thats not that important, a LAMP stack makes more sense. Less scalability, potentially less security but also less administrative overhead and costs.