Just a summary from my experience.
Security
AWS RDS : Give access to trusted people.
Same Instance : Accessible to all working on the project.
Setup
AWS RDS : Much easier. Just few clicks.
Same Instance : Go through docs on setting up one.
Scalability
AWS RDS : Attach it’s own RAM and SSD.
Same Instance : Limited to your server.
Backup
AWS RDS : Backup anytime. Auto snapshot available as well.
Same Instance : Manual backup.
Performance
AWS RDS : Much better since it runs on it’s own server.
Same Instance : Still Good but depends a lot on the traffic at the moment on the server.
Cost
AWS RDS : A bit high. Still, you pay as you go.
Same Instance : None. Only the server cost that you run everything on.
If you are working on a small project and will be having less traffic, then stick to database installed directly on the instance.
Also, Amazon Web Services gives 12 months of free-tier to new users. Might as well just create a new account, verify your details and setup an AWS RDS with EC2.