Bill Belichick v Rex Ryan
Belichick has essentially leaned on Tom Brady's arm completely in order to get production from his offense. He ripped Brady during the season after the offense's early struggles(and Vince Wilfork for the defense). Brady has shown a massive improvement in leadership since that moment and the offense has sparked into life to an insane degree. Belichick's demeanor has remained constant throughout the season(and his career) through the good times and the bad. His calm nature and focus on the job at hand has been vital to keeping a young team from getting ahead of itself.
Belichick is a modest speaker with a powerful mind.
Ryan has been bloating his team's ego since before the season even started with his brash statements during press conferences and ridiculous claims that his team is the best in the league. Ryan is a good football coach, not a great one. He was a great defensive coordinator with the Baltimore Ravens running one of the best units in the league and he has brought that level of performance to the Jets' defense. The Jets are one of the best teams against the run with a less than stellar front seven. Darelle Revis' climb to being the best cornerback in the league was only aided by Ryan's arrival in New York. Defensively the Jets are a brilliant team, potentially the best in the league. Offensively they are closer to the worst.
Ryan has placed his faith in Mark Sanchez as his starting quarterback and the leader of his offense. He is too stubborn to realize that Sanchez is the problem on his offense. They are a solid rushing team but the second year quarterback is a poor passer. Ryan is not an offensive mind, his knowledge of football is purely on the defensive side of the ball.
As a player manager Ryan does not know how to manage ego's, just massage them. Ryan proclaims everything about his team as brilliant because he does not have the ability to actually separate his head from his heart. He is overly boisterous and has a very powerful relationship with the media that has made him an icon as a coach. He may be iconic but as far as coaching goes, Belichick easily bests him.