When someone gives you an API, they're promising you that "if you run code X, Y will happen. When you screen scrape, there's no such promise from the provider, and many providers have items in their terms of service that explicitly forbid screen scraping. From a technical standpoint, this means their page/application may undergo changes that will break your screen scraping, wither accidently or purposefully by the provider. This is why CAPTCHA's exist.
Also, increasingly, these applications are using more and more "AJAX" style architectures, which means you're committing yourself to reverse engineering how their application works, as well as keeping up with the changes each application makes.
Finally, well, you're doing it wrong. Email is a set of protocols in and of itself. Most providers have a way to access email via POP3 and IMAP. I'd look into hacking PHP code to interact with the POP/IMAP servers which, like an API, are a promised set of behaviors. You also have the advantage that code written for one provider will likely work (with minor tweaks) for another.