To me the biggest difference between Holmgren and Carroll is Carroll's flexibility in changing to meet the needs of the modern game. And I think that is a direct result of Carroll's "demotion" to the college ranks.
Holmgren won in GB and was paid more than any other coach in the league at the time to come to Seattle and replicate that success. His position was as a proven NFL winning coach and he tried to replicate his Green Bay days in Seattle: build the offense first through the draft, patch the defense with free agent spackle, and go! Of course he had some questionable draft selections along the way which slowed our progress (and even he admitted later that he should have focused on the defense with draft picks first since it's a quicker repair than the offense). But the expectations of him being an offensive guru (and the hubris that came along with that designation) directed him to keep treading familiar ground instead of trying something new.
Carroll was a coach trying to get back into the NFL after finding success in college, and I think this time around he understands that you have to fit the system to the players, not the other way around. You only get a brief window of opportunity at the NFL level so you better do everything you can to succeed, even if it means thinking outside the box! This flexibility meant fixing the D priority #1 (helps that he's a defensive coach of course), then concentrate on finding a QB and building the O-line to cement the offense.
My opinion of Holmgren has changed only because PC/JS have made it look so easy. At the time I thought Holmgren was doing a good job because he was--compared to the crap we did the decade prior. Only now am I able to see that Holmgren was a below average GM and an above average, but flawed, head coach.
I'm beginning to feel like I always imagined Bill Simmons and the NE fans feel. They went from having zero expectations because they sucked for 30 years to expecting excellence every single year. Looking forward to feeling this way for at least another dozen years thanks to the excellent work this front office has done!
The past is a foreign country: they do things differently there."---LP Hartley