Bret was FAR better for many reasons... you have to take into account some major influencing factors first.
Shawn had in effect 2 careers... his first run and the post 2002 return which was not only better than his first but by many is considered better than Bret's whole career. It's only really fair to compare the first career, because while Bret had a "comeback" it was after a stroke and the concussion so it can't compare. In reality the Shawn who returned was a different man.
You also have to accept some "ugly" facts about Shawn, that his first career is tainted by a near pathological inability to do jobs or drop titles in the "time honored way", he'd lose his smile, get fired/suspended, get beat up by Marines...he would also feign injury a lot. Both adversely affected the stories he would tell in the ring and it culminated in Montreal. If Shawn had done "better business" over the years, Bret would likely have had fewer qualms about losing to him. Even Undertaker had to basically "ensure" Shawn dropped the belt to Austin as instructed after the previous years debacle leading to Final Four.
Bret was a guy who, if he had to lose, could go out there and make the match of a quality that the guy beating him looked incredible, but he always emerged with equal credit. Summerslam 92, the match with Owen at Mania and the double turn at Mania with Austin are his "greatest stories" to most, but go back and look at other matches too... The Survivor Series in 90, most remember it for Taker's debut but the actual meat of the match was the final one on one segment with Bret and DiBiase... it was the match that earmarked Bret for a singles run. He could take that loss, but make it look like he won, same as far back as Mania 4, losing the Battle Royale to Bad News Brown. Shawn could NEVER do that... he had to win the match or had to have an excuse for the loss such as "Jose had a heart attack and I got hit with a camera" or "Tyson made a fast count".
Shawn was VERY good in the ring, but his matches were less about storytelling and more about upstaging others on the card. The nearest he truly got were the matches with Hall for the IC but even then he couldn't do it without the ladder. If given a "lesser opponent" Bret could still tell great stories, his feuds with Lawler had twists and turns like Hakushi and Doink, but the matches told great stories... Even his matches with the Patriot or 1-2-3 Kid. Shawn couldn't do that, he treated non champion opponents like dogshit... even if they were as good or better... look at how he treated Davey at One Night Only... he wasn't willing to tell a story, he just wanted the win, cos he could. In that first title run he had, his opponents did the story telling for him and got the shaft... Even if Davey didn't deserve the win at KOTR or IYH 8 or Foley wasn't champion material... Vader was... Bret would have taken any of those 3, lost the title and made them look immense doing so to the level that you would have been desperate to see the next match.
Shawn had arguably more raw athleticism and ability than Bret, but so much remained untapped due to him not using his gifts the right way and basically being addled on substances and so arrogant/being indulged. A big part of the real life rivalry was that Bret had been forced to fight for what he had, show improvement and consistency over many years in the Tag ranks... whereas Shawn hadn't really done that. As good as the Rocker's were and as much as you can say he and Bret were equals, they weren't... Bret was the "Captain" of the Foundation even as far back as Mania 3... Jannetty was always the same part of the Rockers and the one most expected to move forward... Shawn put paid to it of course but there is this myth that Shawn was always this wrestling god... he wasn't... he was good but like contemporaries like Taker and Nash they improved once it had already been decided they "were gonna be main event". In Shawn's case, Bret played a MAJOR part in elevating them all, teaching them in the ring... Shawn wouldn't have been what he did become without Bret's input throughout his career, the match for the title with the Rockers, the 1992 Survivor Series, Mania 12 and finally Montreal... the final proof is that without those 4 matches, Shawn's career isn't anything... Bret's already was... he'd done the hard work to get the belt and become the best on the way.
So much of Bret's appeal in the ring and out of it was he was the "everyman who made it" rather than the entitled prima donna. That's why his turn on the US fans worked so well, why people hated Vince so much after Montreal and why his latter "bitter old guy" thing was so disappointing.
In flat out technical terms, Bret wins... in shoot terms... Bret wins... in "showmanship" rather than storytelling, Shawn wins, in charisma and personality, Shawn wins... They are almost two sides of a coin, head and tails... but the leader, king, queen etc is always heads... and Bret was the head of the WWF for a lot longer than Shawn was and rightfully so.