Dani, you raise a very important point. A few websites explicitly say you can scan a code off your phone; a few explicitly say you can't, but must have a print out; and many are moot on the subject (at least before you've paid). With more and more things requiring advance tickets (so people aren't shut out) or having financial incentives to get them, and with more people using phones rather than computers to buy these advance tickets, what happens when you can't print them?
I foresee this changing, with more things having the ability to be scanned off the phone, but for now it's a transition time.
If you go to the Tower's ticketing page (the same one as for all the royal palaces), it says the following:
At present we do not accept tickets on mobile devices. If you choose to ‘Collect on site’ (not available for Kensington Palace general admission), the credit/debit card you pay with and confirmation email/voucher will need to be shown when collecting tickets from the Ticket Office before going to the gate.
So if you won't have a printer, it looks like you can book in advance and save the money, but can't save the time of waiting in line.