Unable to connect to a new installation of sql server

I just installed SQL Server on my Windows Server 2012 R2 box. The installation went well but my apps can't connect to it.

If I use the public DNS name for the server in the connection string, the stack trace looks like this:

[Win32Exception (0x80004005): Access is denied]

[SqlException (0x80131904): A network-related or instance-specific error occurred while establishing a connection to SQL Server. The server was not found or was not accessible. Verify that the instance name is correct and that SQL Server is configured to allow remote connections. (provider: Named Pipes Provider, error: 40 - Could not open a connection to SQL Server)]

If I use the server's IP address, this:

[Win32Exception (0x80004005): The network path was not found]

[SqlException (0x80131904): A network-related or instance-specific error occurred while establishing a connection to SQL Server. The server was not found or was not accessible. Verify that the instance name is correct and that SQL Server is configured to allow remote connections. (provider: Named Pipes Provider, error: 40 - Could not open a connection to SQL Server)]

I see its the same error both ways but I've been poking at this for 3 hours and haven't been able to find the problem.

Here's what I've done:

1. I've verified that remote connections are allowed
2. I've verified that ports 1433 and 1434 are open on the firewall for incoming and outgoing traffic (both the Windows Firewall and the AWS Security Group)
3. I've verified that port 1433 is specified for TCP/IP in SQL Server Configuration Manager
4. I've restarted the SQL Server service
5. I've checked netstat -a and there's nothing listening on those ports despite all of the above.
6. I've verified that SQL Server and related services are, in fact, running
7. I've verified that TCP/IP is enabled - this server serves my websites and under Configuration Manager, TCP/IP is marked as enabled.

I don't know what else to do so if anyone has any suggestions...