Yes, you're missing something.

What happens when you click on an input of type submit? The form is submitted.

What happens when you click on a button? I don't know.. depends on the button.. but nothing leads me to think it would have a default behaviour of submitting the form..

Dig deeper into your HTML source. Is the button later bound to an action with javascript? Remember, WWW::Mechanize can't work out what javascript code does, so you're going to have to assist it. If the javascript submits the form, then perhaps you should call $mechanize->submit().