I have found the solution!
I set up a second email account on my server with an obscure name, then, I changed the POP/IMAP settings to use that account for receiving, and left the send/SMTP details unchanged.
This now gives me the desired behaviour. It also gives me a receive only account ( which could also be desirable ).
If you are on iOS4 or below, you can also refer to Hand-E-Food's answer. My instructions will work for iOS5 ( due to the extra checks iOS5 makes to verify what you've entered )
In general, this is probably because one of, or a combination of the following reasons:
1. Outgoing mail server requires authentication
To avoid abuse by spammers, many outgoing mail servers require authentication for relaying messages across the the internet. You should check if the outgoing mail server you are using for this account requires authentication, and if so properly set it up in your account settings. These typically use the same credentials as the ones for checking your mail with the same provider.
2. Outgoing mail server only accepts connections from the same network
Many ISPs only allow connections from hosts which are connected through their own network. If sending mails through your ISP's outgoing mail server works fine from your home Wifi, but stops working over a cellular data connection, or an external Wifi hotspot, this probably is the case.
In this case you can use another outgoing mail server. If you have a Gmail or an iCloud account you can use those outgoing servers for sending mails.
3. Incorrect settings
Double-check the settings for your outgoing mail server as they are specified by your email provider, especially server name, encryption and the port to connect to. If everything looks ok, try to recreate the account. For AOL and other popular services, there are presets for creating a new account which will fill out all the correct server details.
The specific settings for AOL's outgoing mail server are specified here:
SMTP Outgoing Server Address: smtp.aol.com (Use port 587 for standard or 465 for SSL connections)
SMTP Username: YourUsername@aol.com (or @love.com, @games.com, etc.)
SMTP Password: password you use to login to Mail
For a secure connection, check the SSL option for IMAP/POP and TLS for
SMTP in your mail program.
Best Answer
This appears to be a problem with Google's SMTP servers, affecting all clients on all platforms (excluding web) and preventing the sending of email.
This has now been fixed:
https://productforums.google.com/d/msg/gmail/xChsjMRFCUc/-YbEFrt5ODUJ
At the time of writing, Google are aware of this and are working to fix it.
https://productforums.google.com/forum/#!starred/gmail/xChsjMRFCUc