My Android 5.1 phone lets me connect to my computer via MTP or PTP. Are there any advantages of using one over the other for copying files between the phone and computer and visa versa?
The Media Transfer Protocol (MTP) is an extension to the Picture Transfer Protocol (PTP) communications protocol that allows media files to be transferred atomically to and from portable devices. Whereas PTP was designed for downloading photographs from digital cameras, Media Transfer Protocol allows the transfer of music files on digital audio players and media files on portable media players, as well as personal information on personal digital assistants.
You'll be able to see most if not all files with MTP:
...whereas you'll only see media-related folders with PTP: