That is a HUGE advantage.
Plus it's easier for the receiver to catch.
Colin Cowherd said today that the catching part doesn't matter because it's also easier to intercept with a deflated football. Oh brother. 
yeah colin doesn't buy into the receivers having an advantage, but he is a HUGE voice for the "how the fck the Pats have so few fumbles?"
He's been on that train for months now. Marginal players come to NE and never fumble, even in the snow. Colin talks about that a lot.
I think Colin trolls a lot. He will say things like "I don't see why everyone accuses leBron of HGH" when nobody is accusing LeBron, just to get that debate going so people will realize it.
There are probably other ways the balls being a little flat helps. Very easy for Gronk to make those 1 handed fingertip grabs in pouring rain and snow. Very easy for Blount to run into 5 defenders in a rainstorm and keep ball tucked in. And brady himself, in 2011, admitted a flat ball is way easier to throw.
Those little advantages add up... Instead of heading into halftime 13-10, the pats are up 21-3 thanks to insane ball control in poor elements. Then the other team has to throw every time while getting tired, and pats can run way more to run them down and eat clock. The Pats have awesome leads early, and just pound out fumble-free running. Smart.