There's a lot of differences. The biggest one to me is seriousness. Carroll was a goof ball a lot of the time acting like a kid on the side line. That great, but not in football. Coach Mike brings the seriousness and lets get to work mentality to this team which was much needed. Coach Mike is a straight shooter, if practice was bad he'll let you know. Carroll was always beating around the bush and sugar coating stuff which I didn't like at all.
There are many coaching styles that can be effective. Pete's style was very effective, until it wasn't. In the first 2/3 of Pete's tenure, it seemed like a lot of players wanted to come to Seattle because of the environment Pete fostered here.
Vince Lombardi vs John Wooden vs Bobby Knight vs Frosty Westering vs Coach K vs Woody Hayes--now that's a RANGE of styles!. I felt that Lombardi was successful DESPITE his gruff syntax, NOT because of. Same for Bobby Knight. Then you have Wooden and Coach K and Frosty who wouldn't be caught dead throwing a chair across the floor or tripping an opposing player running down the sideline for a TD. My children experienced a variety of coaches and styles, with two daughters playing in college, and one daughter who was recently inducted into her D1 university's Hall of Fame and holds school records for career goals and assists. Their best coaches had a variety of styles, but the best ones CARED about each athlete's success and development, and made well-constructed plans, for practices and activities to prepare each athlete, and the team, to maximize their chances for success. (They also game-planned for opponents, to find and exploit favorable matchups) I'd argue that those things are more important, even at the pro level, than the coach's personal "style", as long as that style is authentic and conveys the coach's integrity.
When the Hawks were winning back-to-back NFC championships and one Super Bowl, Pete's enthusiastic style worked just fine. Pete pranking players or bringing in Will Ferrell as "Greg Olsen" in a team meeting didn't cause losses. Eventually, Pete lost his mojo for a variety of reasons, including misplaced loyalty to underperforming assistants, and Pete underperforming as the "Final Authority" GM after the two SB appearances, and Pete failing to adapt to changes in NFL rules and schemes.
In MM I see a very effective coach with a middle-of-the-road type of syntax, cerebral, direct, effective, conveyed with caring and integrity, and not afraid to tell it like he sees it. I see a coach who doesn't try to influence the GM to go after "shiny objects" and who prioritizes the trenches. I see a coach and GM team that don't overpay for NFL-Average talent. I see a coach who has modern, highly successful, defensive schemes. I see a coach who brought in promising assistants. (e.g. Grubb) I see a coach who has been mentored in a very successful organization and is bringing some of that culture to Seattle.
I'd be OK with MM if he were goofier than Pete as long as we still got all the things I just mentioned. Hell, MM can cry like Dick Vermeil, after every game and even every practice if he wants to. He won't, he'll just be his authentic MM self, and I think it will be very effective.