It's a quite recent evolution in Football, in the past it was common to see teams (Chelsea until recently by example) playing with a midfield where roles are precisely shared between 2 or 3 players: in the case of 2, there's a DM who does the "windshield wiper" role whereas the other plays as a boc-to-box. In the case of 3, there's a real DM feeding another who relays to a third, creative or box-to-box
Double pivot means that there's a two man midfield and that roles aren't really shared. One player is a bit more comfortable defensively and the other, offensively. But the main thing to take into consideration is that both works on their weaknesses at training.
Because modern football can't allow a DM to be rash and untidy in terms of passing, we tend toward more complete players.
It's a bit like the central defense, it's rarer to see an old fashioned partnership between a solid CB, tough defensively etc... and his alter ego who's quicker and reads better the play, who is in charge of the distribution. Nowadays, CB are expected to be at least average in both parts of the play: purely defensively of course but also tactically and in their distribution.
Personally I'm not yet convinced by the double pivot system, especially because modern football is also the emergence of offensive full backs who inbalance the team if nobody except the CB cover them (the axis would be exposed). You expect the two midfielders in a double pivot to work together, cover them mutually ; you can't expect one to stay deep and cover the CBs because in that case, the link between the deep player and the liaison player can't etablish itself. If your double pivot shall stay deep, both players stays deep.
The way England played, the tournament was absolutely decent because they conceided 3 goals. But offensively it was poor because there was no link between Gerrard and Young (players too far on the pitch). Carroll provided to be a target, but with Young and Welbeck who are on ground relays and who likes to be feeded by a midfield (to launch counter attacks, especially Young), they couldn't have the service they needed to produce something decent
Strictly off topic, but do you think we can play with a double pivot role without drogba as effectively as we did last season? The double pivot worked brilliantly for us because Lamps and Mikel played the role perfectly which gave us a great solidity in the defense, but one of the reasons as well that it worked offensively is the relation between Lampard and Drogba imo. If you review the last 10 matches or so, half of our goals started with a long ball from Lampard/Mikel towards the attack. Can we replicate that with Torres instead of Drogba? I have my doubts.