FWIW, the are some additional variables in this equation.
The configuration of the library's public network could block all SMTP (email sent from applications), or just block the sending of email if you haven't setup your mail client to authenticate.
On the email providers end, they could block the IP address of the library's public network, if that network had been used in the past to send a lot of spam through.
Where I live, Cox Cable is our internet provider, and a ton of people use the free @cox.com accounts that come with the service. Cox won't let you send email from any mail application, when you're not connected to their network, unless you do some funky manual setup of your email client. So your notebook on your home Cox connection would send email fine, but the minute you use a free public network, you'd be unable to send. Again, this is a limitation imposed by them, not the email client you're using.
Do any of your email accounts in Mail.app work OK when you're at the public library?
