Great that you provided screenshots, makes it much easier to help :)
It looks to me like your database is in a private subnet, which means there is no route from the internet to your database. The way to fix it is on this AWS page, which in summary says "change your route table to make your private subnet a public subnet by adding a route from the internet to the subnet".
The "allow public access" doesn't magically provide access from the internet to your database. I didn't know what it did now as I have never needed to do that, but this page tells me it simply allocates the RDS instance a public IP. Without a public IP there's no way to connect direct from the internet, it's only accessible from the VPC IP range. After ticking it you still need to provide security groups, routes, etc.
Make sure your security group allows ingress only from your public IP, not from 0.0.0.0/0.