With POP3, you have two options for message handling:
- Download the messages to your device now talking to the server and delete the messages from the server. Obviously this does not work well for multiple devices.
- Download the messages to your device now talking to the server, mark these messages as "downloaded", and leave them on the server for other devices to download.
With POP3, there is no syncing of the message folders, just download and send. Any concept of a trash folder on the server is there for when you delete the messages while browsing them on the server.
With IMAP, your mail folders are synced between the server and device. Moving the message on the phone to a new folder does the same on the server. Ditto for moving the message into the "trash" using the delete command.
I have my phone using the IMAP protocol so that it stays in sync with the server folders.
I have my computer using the Exchange protocol to download all the message folders and remove the messages from the server, which also clears them from my phone on its next sync with the server.