The advantage of the computer is that it makes a good back up if you ever lose a paper logbook. It also makes it much easier to total hours and they keep track of currency for you. Still need the paper logbook though.
During you training keep both a paper logbook and a computer logbook. You'll need a paper logbook for signatures and endorsments. Once your done with your training you can use only the computer logbook. The only flights in my paper logbook since gettting my MEI are a couple checkout flights.
Ditto what everyone else has said, but also think of the required items that a student pilot needs with them in the aircraft during solo flight. Your logbook is one of those items. It's pretty easy to just pull it out and show it to anyone if need be. The proof of pre-solo flight training and all of the endorsements are right there in you logbook.
If you had an electronic logbook on a PDA, what could happen if the batteries went kaput and the FAA wanted to look at your logbook?