What do you like best?
Can store mail locally instead of in cloud (unlike Gmail, Yahoo! Mail, etc.)
Works with multiple mail accounts
Works with POP and IMAP servers
Expandable via "Add-ons" -- many available
Very configurable, especially with add-ons
Full text indexing of emails for fast searches
Available for Windows, OS X, and Linux
What do you dislike?
Full text index breaks too often
Account settings are too complicated
Can't copy account settings from one account to another
Can't duplicate accounts
Can't apply changes to multiple accounts at once
Mail folder selection is done only through drop-down combo boxes, which is a pain for non-trivial structures (this can be somewhat alleviated with add-ons like Nostalgy)
Recommendations to others considering the product
If you previously used Eudora, try the MoreLayouts, Nostalgy, and QuickFolders add-ons.
Some email providers let you access your mail via POP3 instead of IMAP, so it isn't retained on their servers.
You can also set up a local email server (like hMailServer on Windows) to download email from your providers and store it locally while letting you access it via IMAP from multiple computers. To access it from outside your local network use SSH tunneling or a VPN. I use Thunderbird this way.
What business problems are you solving with the product? What benefits have you realized?
I don't like cloud-based email because I prefer to have control of my data. Thunderbird lets me keep my mail -- including customer messages -- on my computer.
I used to use Eudora, but support for it stopped years ago, and it has issues with recent versions of Windows. Thunderbird is my replacement.