I have Thunderbird open all of the time, and I have two Gmail accounts I actively use.
It's very nice and like the others said I prefer to have my mail local if I can. Web based is just a nice feature that's a last resort, IMO.
I adjusted the server settings on thunderbird to re-scan for new emails every minute, so I always know when I get email right away, unlike web based layouts that only show me new email after I refresh the page.