The first email address is what is used for say registering lost password and for board messages sent via email.
This address is not displayed to the public. You can only find the email from the control panel as a admin.
The public email is for if the member wants to have their email viewed and used by others such as spiders,spam bots and other members.
So you don't even need to fill out the public email.
If you look in the user list here. Say for instance Gizmo.
You will see he has a public email address but it is kinda cryptic to keep it from spam scripts sending email to him.
