EC2 instance not responding to ping

Working from the bottom up, layer three would be VPC subnets and routing along with your own local routing. For routing in a VPC, check your subnet routing tables. For an instance to be routable by a public IP address, the instance needs to route to the VPC Internet Gateway which will maintain a 1:1 mapping of the public IP address and the internal private IP address for traffic going and traffic going out of the VPC.

For layer 4 (transport) issues there may be a problem with your security group inbound rules, the VPC Subnet ACL, and lastly local firewalls. Another possibility is an intermediary carrier is dropping ICMP as they apt to do from time-to-time for QoS reasons.

Working from the inside out, if you can ping other sites such as www.google.com and get a response, it's unlikely your network is the problem. If you can't ping www.google.com at all, then the problem is probably very close to you. If you can ping www.google.com, check your security group rules allow ICMP, check VPC Subnet permits ICMP, and also double-check your instance firewall as it's possible you're dropping the ICMP packets with IPTables (or something similar).

That you can SSH is a good sign, because at least we know you can route to your instance.