If I have to guess, you are probably having a different issue that has nothing to do with SQL Backup.
The first thing you need to do is verify that you can connect to the SMTP server from the SQL box. Have you done this with say... Putty or the Telnet command connecting to port 25 on the SMTP server?
http://technet.microsoft.com/en-us/libr ... v=exchg.65
If the test fails, then you need to start your troubleshooting there and take SQL Backup out of the equation.
It could be the Windows Firewall on either server blocking the connection, or a setting on the SMTP service not allowing relaying or a network device in the path not allowing the connection.