You can. But you won't get an advantage because it will confuse the system. In my experience, if I'm logged on at my computer and my phone, when I make a change on my phone it either resets my computer or logs me out completely, or makes it so my requests on the computer don't go through.
I do recommend being logged in to both though, because it's not entirely abnormal for either the app or the website to be completely busted on the day you need it to work the most. When I booked my ADRs a few days ago, for example, the app wouldn't connect to the server, so I had to jump on the computer to make my reservations.