1. Control. A third party can change anything about the service any time they want, and if you don't like the change they made you're screwed.
2. Expectation of privacy. Because I am not contracting with a third party, the government cannot argue that I have waived my right to privacy. (As a practical matter of course this matters not at all. If the government -- or anyone with the right technical skill and access -- wants to read your email they will. But if push ever comes to shove in a court of law it could matter.)
3. Spam filtering. I think the whole industry is doing it wrong. The Right Way to filter spam is to use your outgoing mail as ground truth for what is not spam. I have a custom spam filter that I wrote based on this idea and it works like a charm. No Bayesian analysis needed. I don't even look at content at all. Just the headers are enough to achieve >99% accuracy.
Could you elaborate? Does this mean that email from people/domains you haven't corresponded with before is spam?
I was planning to institute a system where my contact page included a special keyword to include in the subject line to get past the spam filter, but that has turned out not to be necessary so I haven't implemented that yet.
The only remaining case is things like confirmation emails for new accounts, but those just get lumped in with the other cold calls. They are super-easy to spot because I'm almost always expecting them, so they are always at the top of the list.