As socket connection test was successful, but you can't connect with a mysql client, take a look at the output of netstat to see what state the connection is in (replace x.x.x.x with the actual IP address of the RDS instance):
netstat -an | grep x.x.x.x
If you see the 'SYN' state when using a MySQL client, you might be running into an MTU issue.
RDS, at the time this is written, may not support ICMP packets used for PMTUD (https://en.wikipedia.org/wiki/Path_MTU_Discovery#Problems_with_PMTUD). This can be a problem if you're trying to access RDS or RedShift that's in a VPC from a classic ec2 instance via ClassicLink. Try lowering the MTU with the following, then testing again:
sudo ip link show
# take note of the current MTU (likely 1500 or 9001)
sudo ip link set dev eth0 mtu 1400
If the lower MTU worked, be sure to follow up with AWS customer support for help and mention that you are seeing an MTU issue while trying to connect to your RDS instance. This can happen if TCP packets are wrapped with encapsulation for tunneling, resulting in a lower usable MTU for packet data/payload. Lowering the MTU allows the wrapped packets to still fit under the limit.
If it didn't work, set your MTU back to its default and engage AWS support for further troubleshooting.