First off, let's appreciate Sir Patrick Stewart's acting chops, shall we?
That was from Chain of Command Part II and is without a doubt, one of the best scenes in Star Trek and television ever. How does this make Jean-Luc Picard a better character than James T. Kirk? Simple: He's more realistic. Kirk strikes me more as one of those "infallible heroic" characters that were practically a stereotype from the old days of scifi. He was always assured to pull his ass and the Enterprise out of danger and never showed any deep emotion. I mean, when Spock died, he didn't cry. During his eulogy at the funeral, his voice cracked like he was going to, but he didn't. Hell, even Saavik had tears in her eyes and she was a Vulcan! When his son, David, was killed in Star Trek III, there was nothing.
On the flipside, you have Picard, who restrains himself on normal accounts, but does show emotion, nonetheless. There's an episode that takes place after Best of Both Worlds, Family, where Picard visits his brother and his family while recovering from the events of those two episodes. There's a scene near the end where Picard and his brother get into a fight that ends with them sitting on the ground, covered in mud, laughing at themselves. Suddenly, Picard goes from laughing to crying as all the emotions he had pent up inside him about his ordeal finally come out. In Star Trek: Generations, when he receives word that his brother and nephew were killed in a fire, he cries. In First Contact, he showed a vengeful side, that he was willing to do almost anything not only to stop the Borg, but to get a measure of revenge for assimilating him all those years ago.
It's not just crying that makes Picard more realistic, though. At the end of Chain of Command Part II, Picard reveals to Deanna Troi during a therapy session that he had come close to breaking, that if that Cardassian officer hadn't walked in when he did, Picard was going to give in and tell his torturer what he wanted to hear - that there were five lights, not four. Can you imagine James Tiberius Kirk giving in under torture? Me neither.
That's not to say that I don't like Kirk. He's a damn good fighter and leader and if I had to choose which captain I'd want commanding a ship during a battle, I'd go with him and not Picard. However, I'd much rather serve under Picard during exploration and first contact missions than Kirk.
Plus, you gotta love a guy who gets stabbed through the heart and laughs at it.