Saying baseball is important and football is fun is an incredibly individual opinion that means pretty much nothing to anyone outside of yourself. I would argue in the national scope of things the Patriots have usurped the Sox as the identity of the region.
This is the perception within New England, but not outside of it. And it kind of encapsulates the reason the Patriots get hated on so much more than the Red Sox on a national level. There are a lot of reasons, of course, but the constant assertion that they are the cultural face of Boston and/or New England displays an arrogance that permeates the Patriots franchise, local media and even the fans.
As it pertains to the question posed in this thread, we were asked whose absence will we mourn more, and whose legacy will cast a longer shadow. At no point are statistics specifically cited there and for good reason. This is a broader set of questions than that, and for that reason, I have to look at this culturally.
With that framework, it's Papi and it's not close. The Red Sox are far more important to the identity of Boston and New England than the Patriots could ever be. They have a much longer history of being the sports identity of the city and the region for starters. Patriots prominence is a literal blip on the radar in comparison. Until 2004, the entirety of the New England sports fandom persona was wrapped up in the concept of the "lovable loser" because the Red Sox still hadn't broken that pesky curse. It didn't matter that the Patriots had won recently. It was considered a consolation prize for the region. Sports fans in the region were still the lovable losers in 2002, and especially in 2003.
Yes, winning in 2004, and again in 2007 and 2013 changed the culture, but that's the point. It was the Red Sox winning that changed it, not the Pats.
And I've spent a great deal of time in the last half decade outside of New England, so I've seen several regional views on Boston sports and to my experience, the Red Sox are the identity. Whether that's been in New York City, DC, Baltimore, where I live now in Roanoke, it's clearly the Red Sox that people key in on. In the decade that preceded that, I spent more time in New England than outside of it, but still spent a decent amount of time traveling, spending time in Chicago, San Diego, LA, northern CA, trips up and down the east coast and a drive across the northern states and back. Again, always the Red Sox. In fact, the further back you go, the more intensely that is true.
Yeah, you can argue that means the Patriots are closing the gap in recent years, but again... blip on the radar. The Red Sox are objectively, unquestionably more intrinsically tied to the identity of Boston and New England than the Patriots, especially if you get out of the bubble of New England for a bit.
Both Papi and Brady are enormous icons for the region and both have played incredible roles in their franchise's successes but the Red Sox are far more important culturally speaking (now and especially historically) and that gives Papi the edge even before we dive into the many great points made in this thread about how they connect to and relate to the city and the region. He'll be mourned more and his legacy, which includes so much more than just statistics, will stretch further when we look back in a few decades. Brady was the better player, Papi was the bigger icon.