One argument against emails-as-names is that some people change email addresses frequently - unless you have some way of accounting for that (and you definitely wouldn't use the username as the contact field), the username will eventually go stale for some users.
You should allow changing the e-mail address. Of course, changing it requires the same validation, but I'm not really sure why there'd be an argument against it (though I do note many sites/applications don't have a mechanism for this).
At the same time, as a user, you need to consider your own e-mail setup. You should not use your ISP-provided e-mail address whatsoever, there is absolutely no reason to tie your contact to an account you may not keep.
Likewise, you should not be using your school or work-provided addresses for something that you expect to use beyond going to school or having that job.
If you're changing your e-mail address frequently for personal reasons (eg, someone harassing you?), then you should actually consider creating an account you use only for authentication, and a separate one that is your public-facing persona.
Having separate addresses for your contact and sign-ups is actually not a bad idea anyway. It actually makes it a bit harder for someone to break into your account somewhere (because they don't necessarily know what address you used), and you get the freedom to change/abandon your public e-mail address if necessary.
Failure to do so means:
- The user who actually owns that e-mail can't login
- The user who signs up can potentially impersonate the real user (depending on what your app does)
- The user who signs up can't reset their password
- The user who actually owns the e-mail can take ownership of the account (by resetting password)