I don't really remember Valdes at Barcelona, but Neuer was a ball playing keeper before Pep arrived at Bayern. He obviously prefers the ball out short and for the team to move the ball up the pitch together, but that's as much about the whole defence and midfielders working together as it is about the keeper, I can't see why Hart can't play a short pass quickly if he has confidence the players can deal with it and won't give it straight back to him. When people talk about Harts distribution they usually mean his kicking which isn't the best and is usually at shorter players.
The thing with Neuer is that you can always pass the ball to him even under pressure. He is not just a good goalkeeper but could play atleast 2nd or 3rd league as a field player. Like this he is not just the goalkeeper but like an additional field player - and is essential playing the high line Pep prefers (and is needed to keep small distances between the players for the short passing game)