A couple weeks ago on the radio pregame show, Cassidy mentioned (to paraphrase) that he hasn't tried to change a lot of things, but one focus was to get the puck through the neutral zone quicker and attack the other team before they could set up defensively. I think that helps explain some of their success being more creative, getting more dangerous shots, and avoiding some of the perimeter cycling that many have astutely observed. I don't think Claude discouraged getting good scoring chances, but their transition game was not one of their strengths under his regime and it had a ripple effect on their overall offensive game.
I say the following while acknowledging up front that things are rarely black/white but loaded with gray. It seems to me that Cassidy is also enabling the team to play a bit looser or more intuitively and with more confidence while chasing many of the same/similar schematic ideas from Julien's tenure.
We know athletes practice and train to be able to execute their training under pressure, and in a perfect world do so intuitively without their minds overthinking the process. They train that Claude wants me to do X with the puck when Y happens. Swap out coaches, and if the system doesn't materially change - then doing X when Y is still the same. Why better results then? Could be a thousand reasons: from schedule, to luck, to perception, nagging injuries, line pairing adjustments, a new player, alignment of moon/stars, a short term adrenaline bump/kick in the ass because "new coach", etc. But PART of it could be the coach, and how the coach interacts with his players.
There is a psychology to leading/managing people and in this case athletes. There are coaches like Francona who make it known that Pedroia is his second basement no matter how bad it gets (actual truth/realities aside). This takes some amount of pressure off of Pedroia in spite of his growing failure at initially adapting to the majors. Pedroia gets over the hump and would run through a wall for Francona. (I'm NOT saying Pedroia wouldn't have made it for any other coach, just an example.) Other coaches are either unable or unwilling to convey the same message to that player. Perhaps they feel the player is either strong enough to swim on his own or doesn't belong. There may be some truth to that as well. But we know the confidence of the player often is the difference between success/failure - some days flipping day to day, game to game. Some people are subtle at leading, we don't even recognize when they guide us. Some are more sledge hammer-ish when they try to lead (my own personal skill set :/ ). Some players are freed from their nerves when told "you're here for two games, if you go 0-8 you are going back down, if you go 8-8 you are going back down. Go get'em". Some players need more reinforcement, some players need to be challenged when they skid. Each player is different. Sometimes styles just don't mesh, personalities don't click... the locker rooms we are talking about are filled with egos - some fragile and some belonging to divas too. We all know one size doesn't fit all - no matter what the ad execs say.
Professional players seem to come in 3 flavors: 1) Forever blinded by the bright lights and do not belong (or are going to fail based on work ethic, inner demons, etc). They are most often weeded from the system during the journey up the ladder - no matter their physical gifts. 2) Players who are going to succeed because it is in their DNA and it is wired into their minds whether by their parents, their coaches/teachers along the way, or perhaps by God. Players in either of these two groups are not going to be greatly influenced by a Julien or Cassidy in terms of success/fail - it is likely already baked in. 3) The largest group is where a coach makes his leadership bones. When a player isn't playing intuitively he isn't at his best. A player who steps onto the ice knowing that a mistake will get him splinters on the end of the bench isn't playing intuitively, he is playing not to lose. When a player in group 3 believes that his coach doesn't like him (accurately or not) he is unlikely to play at his best. Alternatively, a player who believes his coach will play him through the mistakes as he matures at the highest professional levels (again - accurately or not) is going to have one less mental obstacle keeping him from playing intuitively. When the message is "we demand your success or else" some percentage of players are just not going to get there. When the message is "we demand your success, and here's how we are going to get there together" a larger percentage of athletes might get there. There are lots of messages a coach conveys (it is their job after all), from schematic, to technique, to correct, to praise, and as they tell us in the movies - to inspire. How the message is delivered - and more importantly how it is received can make a large difference - critically so to those in group 3.
There are extraordinary athletes in the "not going to make it" of group 1 above. There are some lesser athletes in the "succeed no matter what" of group 2 above. We all know the biggest skill gap between professional athletes and those stuck on the developmental rungs below them is found in their heads. Some coaches are known for imbuing confidence in their players, some are not. Some players need to be tempered like metals over time in order to get to where the team needs them. Cassidy seems in the short term to be able to do those things, getting the best of the most players. He is a bit past the adrenaline bump or short term ass kicking managerial changes typically bring. I think his approach to player confidence, mixed with continued improvement of player skill set by the GM and this team could be heading in a good direction for the longer road. Admittedly I don't know enough about his Xs and O's to know if he's the right answer long term, but I like his direction.
EDIT: TL;DR - Yeah Cassidy for more than Xs and Os!