When diagnosing RDS instance connectivity issues there's a few layers to peel back.

1. Is the instance using the correct subnets (public for public access, private for almost all other situations)
2. Is the DB configured to be publically accessible if you're using public subnets
3. Does your security group allow the connection you're trying
4. Do you have the correct credentials

The first two are pretty easy to check in the AWS console, but the easiest possible way to test external connectivity is to test with telnet:

telnet my-db-hostname 3306

If you make a successful connection then you've established the following:

• DB is in the correct subnet
• DB is configured to be publically accessible
• DB has correct security group and security group rules

So if all that works, then the only possibility is you've got the wrong credentials. In your situation I think you're using the wrong username root, which should be admin if you're using aurora

p.s: You might want to modify your question and remove the hostname