When do I need to set "Identity / Account" in the client outgoing e-mail setting?

You only have to check this box if the mail server needs outgoing mail authentication or SMTP authentication. You need to check with the administrator to find out if this is the case for the mail server.

When you check this box, and authetication is not necessary, you will not be able to email out. When you forgot to check this box, and the mail server needs authentication, you will also not be able to email out.