Diagnosis

1. You have set up your JIRA application properly (EC2 Windows 2008 Server)
2. RDS installation of MySQL is also complete as per Connecting a JIRA application to MySQL documentation.

Symptoms

1. You get Error connecting to database...Communications link failure The last packet sent successfully to the server was 0 milliseconds ago. The driver has not received any packets from the server.
2. If you try to connect to RDS you are connected without any issues, but JIRA application cannot connect.

Workaround

It seems you need to configure the RDS security groups to accept remote connections specifically, including the JIRA applications instance.

Check this page for details.