Connecting from SSMS to an EC2 SQL Instance

If the server is installed as the default instance, you should be able to connect with DNS name or IP Address. The "\SQL2008" you've added to the address would be used to connect to a named instance called "SQL2008".

Try specify the port in your connection string e.g. "123.123.123.123,1433". Without the port specified, the client will query port 1434 initially to determine which port to use. Alternatively you could open port 1434.

SQL Server Browser Service

When SQL Server clients request SQL Server resources, the client network library sends a UDP message to the server using port 1434. SQL Server Browser responds with the TCP/IP port or named pipe of the requested instance. The network library on the client application then completes the connection by sending a request to the server using the port or named pipe of the desired instance.

Edit: If you can connect using MACHINENAME\SQL2008 when you are RDP'd to the server, you must be using a named instance. If this is the case, your connection string with the port specified will be "123.123.123.123\SQL2008,1433".