The difference between 7 HS and 6 HS is quite simpily the opening turn of the game.
More often than not in a deck fronted by a 6 HSer, i will not draw enough or the correct foundations, or there will not be enough versitality in my hand to allow me to play safe foudnations (once that cannot fail of my decks own accord) So very many times, even with a properly curved deck with a very strong spread of foundations difficutly, ill draw an opening hand for a 6hser that is like 1 asset, 2 attacks, and 3 foundations (each with a 2 difficulty). By the time I make the 3rd check to play that 3rd foundation, I can and sometimes will fail it by rolling an attack. 7hsers are much more versatile against this
The other part of the opening turn equation is that with more cards in hand first turn, you can play more cards first turn. I've have decks that could play out 7 cards turn one. a 6 HSer will never be able to do that. Being able to play 5 foundations and still have the 2 cards in your hand for The anti K or Antisocial are a big deal as well. Even if they dont make all thier checks, a 7hs will be more able to see more cards from thier deck and draw more cards come turn 2, without having to devote aquakinesis to it (also aqua may help 6 HSers but it helps 7 HSers eqally as well, if not better because they are more equipped to use the cards they draw because they are better off to spam turn one and pass the checks for more subsequent cards).
I feel like the fact that most 7HS characters lend themselves to aggro is also something of note, because most aggro better early on than the 6hs defence based characters can defend themselves.