Patrick Stewart played Gurney Halleck in the David lynch making of Dune, by the way.
Stewart's has almost exclusively played the authority figure role. He is excellent at what he does, but he is more than a bit type cast and no roles come to mind where he plays against his natural type. He has a supurb command presence with voice and athletic physique, so I guess he is doomed to play our vision of a proper king or leader of men.
Shatner, on the other hand, hasn't quite got that voice or physique. His attempt at command presence comes off as more of a street swagger. Exactly what Rodenberry probably wanted at the time. Pitiful, puny, glorious humanity against the unknown. Kirk is an American of Scottish or Scots-Irish descent with just a bit to prove to himself, the federation, and the klingons. He's got to hold things together with force of will when everything else fails. And, while athletic, his presence is not so authoritative. Which means the audiance may have some doubt about the outcome. Kirk's galaxy isn't quite as tame as Picard's, either(the borg are the exception). Therefore, Kirk is less diplomatic and faster to draw his weapon because of that wildness. It is the cold war period in the real world afterall! Kirk is a bit over the top - that fit during the sixties. Kirk values desiciveness and isn't going to loose sleep over might have beens. He is not George Bush, however, in that he seeks out councel from all quarters and allows vigorous and open debate.
Picard comes along during perestroika with the Soviets and the Klingons - The federation is in a much more secure galaxy - starship captains are diplomats and restaint is required.
Stewart really does project the presence of a special individual that you will look to for an almost parental protection and guidance. Seamingly 10 feet tall and bullet proof with the voice of God he has a much more mature resolve and reserve demeanor.
Kirk is around thirty-five when he is captain of the Enterprise. Picard is probably around fifty. A good bit of the difference in thier styles of leadership could be attributed to thier age differences and experience. Kirk wasn't loosing sleep over the might have beens at thirty-fifty. Picard is Kirk years later when his minds starts to revisit the might haves beens of an earlier time. He will loose some sleep over the might have beens, but he has a ship to run in the end.
He has already proven himself and has the wisdom to know that as he solves todays problems many consequences will play out into new problems in the future.