The biggest weakness of both cultures (besides their tendency to piss off every neighbouring race with their arrogant aggression and ruthless treatment of other populations) is their apparent lack of capacity for racial unity. Backstabbing power plays and vicious competition between bloodlines, power blocs and individuals are the norm in both Cardassian and Centauri societies, and this endemic in-fighting weakens their ability to wield the influence they otherwise could. Too much of their scheming is channeled into petty retributions and jostling for position within their own hierarchies, rather than the cause of bettering their people.
The Cardassians call themselves a Union, drawing together in mutual discipline and sacrifice to ensure their shared survival, but in practice they're still at each other's throats, seemingly unable to play nice or put the needs of their population above their own ambition. The Centauri are perhaps more honest about their nature but less sympathetic, having a society dedicated to flaunting resources and power for their own sake rather than as a means to survival. That said, they have a system wherein people are supposed to accept their position with grace (nobility loyal to the emperor, lesser nobles respectful of greater houses, commoners and slaves accepting their place in the scheme of things), but in practice they're completely unable to live in the sedentary and structured system they've created, and behind the scenes they're desperately grabbing for power and influence wherever they can.
Both, I think, have desperately imposed a static, repressive and rigid social structure over their peoples (Cardassians a conformist militant police state, Centauri a class-based hierarchy), and insisted that duty to this system be among a citizen's higest values. In truth, they've both slapped this system over a viciously competitive mindset that is unable to overcome the urge to knock others down in order to climb higher yourself, and it always shows through.
As for who "wins" in a fight, though?
The Centauri are ruthless, but they're also hedonists. They delight in decadence, ostentatious display of impracticality, pointless tradition for the sake of tradition. Excess and waste is practically a requirement for their nobility, and their celebrations are based on an "eat, drink and be merry, because, hey, we're alive and that's great!" mentality. And with the defeat of the Xon, they triumphed over the only real competitor they ever knew. So they believe it's now their birthright to be on top. They're apex predators, at least as I see them, and like all predators they don't expend energy unless they feel they need to. If their status as top dog is threatened, then they "need" to, and they're ruthless and driven (no doubt falling back into Xon war thinking) until the opponent is crushed. That military thoroughness G'Kar speaks of seems a display of precision and power that points to real dedication...but when the threat is over, then it's time to party again.
Cardassians, though, never party. They believe the supposed needs of Cardassia are justification for militant acquisition of resources, and the ruling parties justify their position through military service history rather than birthright. They're constantly pressured into continuing their militancy, either by the genuine desire to stave off scarcity or the knowledge that they hold on to their power only by seeming to serve the needs of Cardassia. Where Centauri are dedicated and ruthless when they feel their power is threatened or else feel the need to flaunt it and remind everyone who's boss, the Cardassians are dedicated and ruthless pretty much all the time, because they constantly have their backs to the wall, partly due to factors beyond their control, partly due to their own messed-up society and its expectations.
So what it comes down to, I think, is: who has the greater
need to win? The old lion fighting to retain leadership of his pride, who would otherwise be lounging about but is NOT going to let anyone challenge him and still knows how to show everyone else who's boss, or the lean, starving lion whose life has been one long battle for survival and who is driven to take that leadership position no matter what?
Who wants it more?
PS: In physical battle, the Centauri presumably use their High Jump Kick to try and hit the weak point on the Cardassian's chin. Whether it works or merely makes them look silly, I leave to more elevated minds to ponder.