You can use POP or IMAP for either one. Here is the difference. With IMAP mail, when messages are downloaded to your e-mail client's Inbox, they also remain on the server. In other words, if you shut off your computer, went to the library, and opened your g-mail account in webmail, you would see all your messages that were previously downloaded to your inbox at home.
Now, let's say that you decided, while on web-mail, to delete some of those messages in the inbox. When you go back home, open your e-mail client, the client will reload the messages from the inbox, but the ones you deleted from the inbox while on the webmail interface will now be gone from the inbox in Thunderbird as well.
Now, if you are using POP, the messages downloaded to your Inbox will no longer be in the Inbox on the webmail interface. Now here is what is rather interesting with g-mail in contrast to some other e-mail providers. All messages that were downloaded to your computer you can still find in the trash box on g-mail's webmail interface. With other e-mail providers, those messages would no longer be on the server at all.
Going back to I-MAP. If you are using an e-mail client, if you move messages from your inbox to local folders, with most e-mail providers, those messages would now be gone from the provider's servers. With g-mail, those messages will appear in Trash. If you move messages to folders you create under the e-mail account, those messages will then appear in the same folders on the web-mail interface. This can be handy if you need to access your e-mail from various computers. If you create folders while using the webmail interface and you move messages into them, you will need to subscribe to those folders in your e-mail client to get them on your e-mail client. With IMAP, generally, what you move in the client is moved on the server and what you move using webmail will be reflected in the webmail client.
The exception is local folders. Those folders are ONLY on one specific computer.
Have I just made things muddier for you?