What does Synchronize mean in Mail.app? In Mail.app, what does Synchronize actually do?  How is it different than "Get New Mail?"
Mail talks to the mail server to show you messages. Only IMAP accounts can be synchronized as that is a formal way for the server and the mail client on your mac to double check all messages to make sure everything is in sync between the two.

Get mail focuses on the Inbox and relies on the Mail app having a "correct" view matching the data on the mail server. 

Synchronize allows for noise/errors/lost messages, incorrect statuses and other real life bumps and errors. It basically double counts and double checks each and every message to make sure things are 100% correct. Synchronize also checks all folders recursively to get everything accounted for - not just the subset of folders that are marked as having new mail.

Even more "deep cleaning" would be if you told mail to rebuild the account or the mailbox. That means that the local storage that mail uses on your mac has problems getting mail or synchronizing. It's the last step before deleting the account and starting over to download things cleanly from the server.

In a nutshell - synchronize should normally do nothing - but it will fill in any gaps as needed. (of course it will get new mail as part of a total synchronization - but it's going to be a lot slower and use more resources on the mac than just asking for new mail)