I use thunderbird as well, and multiple email accounts. I also use the web clients you mention. What Thunderbird gives me, is a single place for all my email accounts and emails I have received from all of them. If I delete an email from gmail, and Thunderbird downloaded it, it stays in there. It is like a big archive for all my email and a central repository to go to for searching across all email accounts (it is easy to forget which account you dealt with a subject on). My iPhone mail app is unable to look back very far, and each gmail account can only look at itself via the website, so this searching through all my emails comes in pretty handy.