I am going to break this down into a few categories because these are two of not just my top 5 favorites, they would be two of my top 5 in an all time best (being as objective as I could be).
Top MatchesJust taking my top 100 right now, Hansen currently has 7 matches to Bryan's 4. You have to get through 4 Hansen matches near the top of the list before you run into a Bryan match. There may be some movement on this front in the future, both near and distant. Some of my more recent inclinations have tended a bit toward some of newer wresting gaining some ground and Bryan's best tends to be what I think is the best of that lot for the most part. On top of that, I generally think we don't really know how good something is until we have had it for 15 years or so. We are just now to that point with Bryan's early classics and I think they hold up really really well. Even still, the point is clear to me. At this point, Hansen's best matches top not just Bryan's, but just about everyone's (sans Kobashi).
Winner - Hansen
VersatilityI think both men actual betray a lot of versatility throughout their career, but Hansen's best matches, I think, tend to exist within a narrower spectrum. Whats more, what he brings to those matches is generally similar. To me, the 3 Kobashi matches, the Kawada match, the bullrope match with Colon, the Andre Match and 88 real world tag finals are his top matches and while those are not one note, its his controlled chaos and bulldozer brawling that more often than not brings a match to the next level. Bryan, on the other hand, is always flexing his muscles in this regard. He is obviously a technical master, but he works as an ace, an underdog, a stiker, and a grappler. He even flashes some brawling chops here and there. Bryan is a master of working to his opponent to get the most out of a match and Hansen was a master of bringing his opponent into his world to get the most out of a match.
Winner - Bryan
Consistency
Hansen's style and his willingness to impose it on just about anyone makes him a relatively consistent performer. There are few people I can bang on enjoying their work in the same way I enjoy Hansen's. Bryan is one of them. I think Bryan is one of the most consistent workers and his ability to be versatile and work with anyone and around whatever situation make him not just consistent, but bankable asset when it comes to greatness. Bryan rarely under-performs, rarely disappointments. Hansen sometimes doesn't quite get the most out of the situation. I know some people like the Misawa matches and I wouldn't say they are bad, but they don't hit the level I would expect given the pieces. I would sort of say the same thing about his clashes with Vader, famous as they might be. Bryan on the other hand exceeded expectations with guys like HHH, Randy Orton, Bray Wyatt, etc. There isn't anyone in the world I would rather bet on to have a 4 star match with any given wrestler than Bryan.
Winner - Bryan
Charisma/"It" Factor
Bryan is about as relatable and likable as they come. Few people are more universally loved than him. Whats more, he won the WWE crowd and the wrestling world more generally over primarily with his ring work. It wasn't that the average fan was enamored with his technical prowess; it was the it factor that he demonstrated every time he went to the ring. He has a knack for making you care about every move. But it isn't really a natural presence. He doesn't feel larger than life when you are in the room with him (I've been in non-wrestling rooms with him and Styles, for example and Styles has more presence). He exudes a quiet confidence that draws the eye, but it is his ability to tell the story that constitutes his in ring charisma and what might be his "it" factor. Hansen on the other hand is a force of nature. He is a tornado personified, a super hero/villain. He is a man who fills a room with his presence and can make any fight feel like a big fight. Ultimately, their charisma is different and - to me - nearly equally impressive. However, if we are thinking of who truly brings something intangible to the ring, I think Hansen edges this out by the very thinnest of margins. His highest highs are better than Bryan's to me not because he is better at any single thing... but they have an it factor that makes this mesmerizing.
Winner - Hansen
Promos
This one might be kind of an apples and oranges situation. What each was/is asked to do is quite different. What is that old saying? "I can't convince you that wrestling is real but I can convince you that I am..." Hansen could do that not just in the ring but on the mic. No one wants to fight Stan Hansen because he looks and talks like a crazy person. While I am sure he had some legit crazy in him, it also appears that through much of his career that he was a pretty level headed mature guy, so some charicter works is on display here. I think we probably have a lot more in terms of promos from Bryan, which is both good and bad. On one hand, we see that he is no probably the best promo in the WWE (at least in my humble opinion). On the other, we have some promos from him that were quite awkward. They were still relatively endearing because.... ya know... Bryan, but they weren't great. Ultimately, I haven't run into and/or don't remember enough Hansen promos to do a proper comparison, but I don't think he ever flexed the peak promo muscle that Bryan has. Bryan can fix the incompetent and unfocussed writing of the WWE in a single night, that says something.
Winner - Bryan
Elevating Others
I sort of covered this already in versatility, but this one is still deceptively close to me. Here is how I see it. Hansen could - if the chemistry was right - take a great wrestler and put on an all time great match. He did this such that at least two of his matches I have legitimately considered for #1 all time. Bryan can elevate anyone beyond what they are normally able to produce, but hasn't yet brought someone into that #1 consideration. That is, whereas Hansen can make great wrestlers other worldly, Bryan feels like he raises all ships a bit more. Again, just looking at my list, both get the best out of quite a few wrestlers, but Bryan probably has the worst dance partner in a top 100 match (Nigel). I think if I really looked back at maybe my top 20 or so for each (which I just skimmed), Bryan would be elevating lesser talent, Hansen would help great wrestlers reach their true peaks. That said, I think one of the big criticism of Hansen is that he could gobble people up and I sort of agree with it. It was a different time and they faced different demands, but I can't think of anyone who was worse off because they got in the ring with Bryan.
Winner - Bryan
Offense
As I mentioned above, Hansen is a force of nature. He brings the firepower like few others. He is big and strong and half blind and stiff. He probably doesn't get quite enough credit for being pretty solid when working on the mat and being able to really construct his offense in a smart way. Looking at the Kobashi matches for example show that he is able to deliver a lot of offense in different, fantastic ways. Bryan gets all the credit in the world for being able to put matches together and package his offense effectively, but doesn't have the punch to his move-set that Kobashi does. I would argue his nuance and versatility actually bring him up to Hansen's level on this though. Strikes, submissions, flying... everything looks good from Bryan and everything is snug and - by all accounts - safe. This one is tough, but I would have to call it a draw. Hansen is higher impact. Bryan is more diverse and creative.
Winner - Draw
Selling
It should be clear enough by this point, but I am forgoing an "in ring" category in favor of bits and pieces that might make up such a category. I have always thought that Hansen's selling was a bit underrated. I don't think the general consensus is that he is bad or anything, but I think his selling is one of the strong points of his work. Perhaps I am not paying attention to the right people in this regard. Going along with the criticism that he gobbles people up, some might see him no selling something and knock him for it, but I always felt like his selling made perfect sense for him and produced massive payoffs long run. Hansen sold strategically and when he really really sold for someone he put them over big. He looked legitimately vulnerable against not just the likes of Andre and Kobashi, but against Kawada and Colon. I think Hansen is a great seller and when he starts bouncing around and screaming it really adds to the story they are telling. Bryan is also a great seller and can do everything you want from a wrestler when selling. However, his greatest strengths as a wrestlers are elsewhere to me. This is another razor close one, but I'd give the nod here to Hansen because I think his selling brings a little more the table at the end of the day.
Winner - Hansen
Rewatch Value
This might be the thing I value most when evaluating a wrestler on the whole. Its one thing to grab my attention in the moment (and there is exceptional value in that. However, it is another thing all together to produce something I want to go back and rewatch. These are two I would consider potential deserted island wrestlers. If i could only watch one wrestler ever again, I would seriously consider both. Hansen is perhaps the ultimate comfort food wrestler. Lots of matches that are just a blast, they are self evidently fun. At the same time, I probably don't have as much emotional investment in any wrestler as I do Bryan. He is one of the reasons I got back into wrestling in college and then really leaned into my fandom again in graduate school. He has been my favorite or one of my 2-3 favorite wrestlers since 2003. I was at Mania 30. He just connected with me in a way that is tough to describe (but most fans feel). Unfortunately, Hansen can't produce that because I wasn't watching live during his prime. I was too young or not alive. It could change at any moment, but if you put a gun to my head today and asked me who I would rather watch if I could only watch one the rest of my life I would probably have to say Bryan
Winner - Bryan
Bonus: Elliott's Question (Brody v Nigel)
This is a little like the last category, but specialized. It isn't just responding to Elliott though. These are probably the two wrestlers most associated with Hansen and Bryan (or at least really close). Nigel may not be everyone's cup of tea, but I still prefer him to Brody, who I have never really cared for (regardless of where the pendulum is on him with general fandom). I am not sure either was a particularly difficult task because I think Hansen accented Brody in a similar way that Bryan accented Nigel. Still, I'll say Hansen's success in this regard is more impressive.
Winner - Hansen
Results
Bryan - 5
Hansen - 3.5
Draw - 1
It isn't an exact science and this isn't an exhaustive list of things to consider (for example, I am forgoing "longevity" because Bryan is still active), but as it stands now I'd have to tilt towards Bryan by the slimmest of margins. What this sort of boils down to me is peak vs full career, which now strikes me as ironic since I generally identify as a peak fan and defend that way of thinking about wrestling. Hansen's highs are about as high as it gets. He has more 5 start matches in my list and more 4.75 star matches. However, when I start to move to 4 and above the pendulum starts to swing towards Bryan. His output is so consistently high to me and he makes everyone around him better. Once again, there is nothing scientific about that. I haven't watched, let alone rated, all of their matches so there is a great deal missing. However, these two are at or near the top of the list of my most watched wrestlers and my gut feeling on these guys resonates with the peak vs full career framing.