Promo: Ric Flair takes the edge here. However, I think that Bret was underrated in this category. Ric was obviously more well-spoken, dynamic, charismatic, and amusing on the mic than Bret. However, as much as I love Ric Flair, he often came across as a bit of a cartoon character (as do many of the best promo-artists of today, such as the Rock). Whereas Bret Hart made it feel real; a task sorely unappreciated even in the latter days of kayfabe. I would much rather watch a standalone Ric Flair promo than Bret Hart promo based on pure entertainment value, but back in the day nobody could get me interested in a match or a feud like Bret. Bret came across as a genuine no-nonsense tough-guy, and he managed to infuse his matches and feuds with a sense of realism and gravity not only in the ring, but also in his promos. Maybe it was because Bret sounded more like an athlete than an entertainer, and unlike many other wrestlers who found themselves in that position he didn't attempt to hide it. Rather Bret stuck to simple, straightforward promos which achieved their goals - putting over the match, the opponent, and himself - and delivered them competently and confidently. That confidence and seriousness worked for Bret and made him believable and intense, whereas many wrestlers who have tried to do the same have come across as boring and/or fake. Obviously Bret can't compare with Flair's raw, natural charisma on the mic (Flair could hold an audience captive with his words in a way that few others have ever done), but Bret did well with the skills he had and didn't expose his weaknesses as often as he might have. So while I've got to go with Flair in this category, I've got to give Bret some credit as well.
In-Ring Ability: Bret Hart takes this. Flair was great, but Bret may have been the greatest (IMO, it's up between him, Angle, or Benoit). It's very hard to criticize the in-ring skills of Ric-Flair, as he was pretty damned amazing. However, as with his promos, Ric did tend to come off as a cartoon character (which can be a good or a bad thing, depending on your perspective), as characterized by his face-flops, top-rope mishaps, and overuse of low-blows and eye pokes. And as others have mentioned, Ric was slightly repetitive. Mind you, those are very minor criticisms which I point out only in contrast to Bret's abilities; Ric stands head and shoulders above the majority of wrestlers in terms of in-ring-ability. In the ring, however, Bret Hart was pure perfection. He simply did not make mistakes. And as as storyteller (ring psychologist), he was unrivaled. Bret elevated pro-wrestling as a storytelling medium to an art-form.
Better Career: I've got to go with Flair on this one. While I could argue that Bret was top-dog in the biggest show in the land more times than Ric, Ric's career was simply incredible. Few wrestlers have had more championships, more accolades, and more respect than Ric Flair. Bret may have upset Ric in this category if his career hadn't tragically been cut short, but it does no good to speculate on maybe's. Ric has done it all, been everywhere, and is still helping to put over young wrestlers today. It's impossible to think of Ric without thinking of his epic sendoff from the WWE, basking in the praise of an entire generation of young stars that he helped to influence. Whereas it's impossible not to think of the unfortunate circumstances surrounding the last few years of Bret's active career: The Montreal screwjob, the death of his brother Owen, the Goldberg incident, and later his stroke. This is no slight against Bret; his career was legendary, and in recent years Bret has been able to reclaim his WWE legacy and put a positive spin on it. However, Ric Flair has simply done more, and for longer, than Bret Hart.
Personal Favorite: Undoubtedly the Hitman. I've always been more interested in the 'sports' than in the the 'entertainment' aspect, and in my opinion Bret Hart is simply the finest wrestler to ever grace the squared circle. And Bret was also in my favorite match of all time (the Iron-man match against Shawn Michael's). And whereas I've always loved Ric Flair's flamboyant and over-the-top persona, I never really identified with it to the same degree as I did Bret's character. Bret Hart's integrity, self-confidence, dedication, toughness, and perhaps most importantly his believability made his character one of my childhood heroes. To this day, he remains my favorite wrestler of all time.