[Warning: Spoilers for Asura's Wrath and the God of War series below.]
I've never liked Kratos much, mostly because I'm not as impressed with him as God of War clearly wants me to be. Sure, it looks spectacular when he kills a Titan by jamming a two-story spike through its chin. But I can't enjoy the show much, because I'm still stuck on Kratos smashing some poor translator's head in for the crime of knowing how to read. I don't remember why Kratos is doing what he's doing in those moments of needless brutality, and he never seems to learn anything, despite the fact that every misery in his life could have been avoided if he wasn't a bastard from the start.
I could just shrug and say that's the nature of the beast, that the power fantasy that God of War is trying to create needs a ruthless killer ready to drop a beatdown at any second. But then I remember Asura, the muscle-bound protagonist of Asura's Wrath. Despite being just a destructive, vengeful, and, well, wrathful as Kratos, Asura also manages to be a stand-up guy who I empathize with and want to see succeed. He's both vengeful and selfless, incredibly strong but not afraid to rely on the strength of others. And he makes it really hard to cut Kratos any slack, because Asura proves that a successful power fantasy doesn't require an unquenchable bloodlust and alienating attitude.
That difference isn't obvious if you just study the bare facts of Kratos and Asura's lives, which look nearly identical. Both were betrayed by those they trusted, leading to the destruction of their families, after which they dedicated their lives to taking revenge. Both rely on unchecked rage for their strength, perform incredible feats of destruction, and end their enemies in grotesque ways. The big difference is that, while unbridled wrath is kind of Asura's thing, he's remarkably good at showing restraint against people who aren't the target of his anger. He's constantly furious with his fellow demigods for treating humans as expendable, and will happily take the brunt of a powerful attack to protect a comrade. Revenge might be the first thing on Asura's mind most of the time, but we're never given the impression that his personal feud is more important that the lives of the people around him. That perspective makes it a lot easier to cheer him on, as he holds himself to the kind of standard of behavior worth respecting.
Kratos, meanwhile, shows no remorse when he causes harm to people who never hurt him, whether he's burning a man alive or making the world uninhabitable by destroying the gods that tend it. In his mind, his actions are always justified regardless of who they hurt, because his revenge is more important than mankind's suffering. Seriously, he says that.
That would be forgivable if Kratos ever realized that his own ruthless behavior is what caused his downfall in the first place: he was happy murdering his way across the world without regard for who he hurt, and Ares took advantage of that by putting his wife and daughter in front of him during a rampage. If Kratos took a moment to consider his own part in his misery and became more discerning about who he hurts, it would be easier to respect him. Instead he keeps being a murderous asshole to everyone he meets, and never acknowledges that anything is his fault, so it's hard to sympathize with his plight when he doesn't care about the collateral damage he's caused.
Those attitudes carry over to scenes that are meant to humanize Asura and Kratos, with predictable results. They each experience flashbacks about their families, both dead in Kratos' case and dead/kidnapped in Asura's. The difference in screen time is obvious: God of War shows a half-hour of violence for every minute of reminiscence, while Asura brings up his daughter Mithra at every possible opportunity.
That makes Kratos' family feel like an excuse for him to commit crimes against god-kind, rather than the reason for his quest. God of War wants Kratos to be a Badass(tm) at all times; it's why you get scenes of him murdering giant monsters, juxtaposed with emotionally-devoid sex scenes that stand as a testament to Kratos’ manhood. He spends a lot more time killing than he does caring, because Kratos is The Man, and that makes it really hard to care in those convenient moments when his family comes up.
Even when the game tries to make him seem compassionate by having him interact with his ghostly daughter in Chains of Olympus and Pandora in God of War 3 (only after realizing Pandora will be useful to him, though), it feels fake given his constant brutality. That woman you crushed to death to hold a door open is someone's daughter too, Kratos. [NSFW for toplessness and violence.]
For Asura, saving his still-living daughter is far more important than taking revenge. Thinking of her (which he does in at least half of the game's cutscenes) pushes him on and stops him from giving into his rage the way Kratos does. Plus, Asura's Wrath doesn't just use his family as a license to let him run loose before forgetting about them. The game takes frequent breaks from the action to focus on scenes of Asura with his family, and show that despite being an unstoppable demigod, Asura's greatest fear is being a lousy father to Mithra. Thinking of his daughter suffering at the hands of the Seven Deities gives him the strength to climb out of the land of the dead, and he reaches the height of his power after a human girl who closely resembles Mithra is killed by a godly rival.
While revenge is part of what drives him, it's secondary to protecting his daughter and feeling worthless for letting her come to harm. Miraculously, those emotions don't take away from the fact that he can kill a planet-sized god by punching the guy's finger really hard. Because the game isn't afraid to show its big, tough protagonist having emotions, Asura feels more genuine and likeable. Unlike Kratos, it's easy to root for him, because we know he really means it when he says he wants to take his daughter back. He isn't just in it for destruction's sake.
When all their attributes are stacked against each other and points are tallied, Kratos' biggest downfall is that he's painfully one-note. He's a marketing slogan given form, the stereotypical ideal of Manliness who falls apart as a character the moment you don't buy into that idea anymore. Asura, on the other hand, is a fully realized protagonist: he has all the strength, testosterone, and manly rage that Kratos does, with a detailed personality as its foundation. That makes him more likeable, but more importantly, it makes him a good character. That alone puts him miles ahead of Kratos.
Plus, Asura would totally win in a fistfight. Let's be real.