It is likely that your ISP blocks the SMTP port 25, most do. In order to get around this they allow you to relay off of their SMTP server. The problem is that this SMTP server usually cannot be contacted on another network.

Generaly it is best to see if your mail provider allows traffic to their SMTP server on a alternate non blocked port (Eg. port 587) that way you are not relaying off of your ISP's SMTP server.