Maybe I just read into things too deeply, but most of these are hard to determine when you really look at it...
Music: This is basically the first half of the 2000s vs. the back half of the 2010s. The late 2000s was the most trash era of all-time, in my opinion, but the early 2000s was such a classic period, and maybe my favorite in my own lifetime. The second half of the 2010s gave us widespread access to streaming services, and thus opened up the world of music to where you could discover unlimited fire. This is the closest category to me, but I would have to edge it to the 2000s on the strength of the several undeniable classics that came out in Neo-Soul and Hip-Hop.
Films: I gotta say the 2010s, because of Netflix and other services. 2000s movies weren't enough to combat this.
TV: I wanna say 2010s for the variety, but these huge shows with annoying ass fanbases (i.e., Game of Thrones, Walking Dead) really make me want to go against the 2010s. I'll have to say the 2000s. It had Chappelle's Show, prime Disney Channel, prime Cartoon Network & Adult Swim, and still had cool music shows like Rap City and the original 106 & Park.
Sports: I don't really watch sports that much anymore because its always been rigged, and I have better shyt to do. HOWEVER, I can objectively say that sports are more entertaining in the 2010s. We've seen Lebron for an entire decade, Black QBs are taking over football even more than the early '00s., and the internet has made sports conversation more entertaining.
Video Games: I stopped gaming years ago, so I'm not the best person to ask this. I personally have nostalgia about Grand Theft Auto and the old Maddens, but once again, I can't speak on today's games. Maybe they're better, I wouldn't know. However, based on my limited knowledge, I'd have to go with the 2000s.
Fashion: This is another tough one. The "popular" fashion was definitely better in the 2000s, but the 2010s are more open, which I love. By open, I mean that not everyone has to have the same style. You look at any rappers or young celebs nowadays, and you see all different styles of fashion. That wasn't really the case in the 2000s. For that reason, I have to edge it to the 2010s.
Politics: neither; fukk politics
Sex Symbols: This is the easiest on for me. I'm taking the 2000s, easily. Every bytch now has the same generic look. The 2000s had women with true sex appeal.
Cultural Shift: Another tough one... The 2000s saw the mass proliferation and ubiquity of the internet. However, the 2010s gave us the complete smartphone/app takeover. The 2000s had a huge shift in the music industry, due to "pirating" and labels/artists not making albums like they used to, because of that. The 2010s sought to rectify this problem with streaming and other online forms of paid consumption, and made music way more accessible. I don't think we'll truly know the cultural impact of the 2010s until a few years from now, but the 2000s laid the foundation for everything, tech-wise. I gotta say this one is a TIE