Great thread, CentaurSuperman! I've actually thought of matching these two up because one or the other would make or miss my top ten films of 2015 list, so this thread is right up my alley. Pardon the wall of text below.
TL;DR - Although I personally enjoy AoU more than TFA, AoU was a disappointment compared to the rest of Phase 2 and Avengers 1, whereas the first viewing of TFA left me wanting to see the future adventures of Rey, Poe, and especially Finn against the First Order, plus TFA was a superior movie from a filmmaking perspective to boot.
I'm going to break it down by category:
- Music: John Williams' legendary Star Wars scores - even if somewhat too familiar - outdo Danny Elfman getting hired by Whedon on a moment's notice to finish AoU's music off when Alan Silvestri left. Also, the classic blaster shot, lightsaber, and space flying and combat sound effects of Star Wars are always excellent.
TFA = 1; AoU = 0.
- Visuals: AoU in some of the action beats with the Ultron bots looked choppy and like a video game. My wife and I even commented on it IN THE THEATER watching the movie for the first time. Force Awakens has some of the "ya, that's a CGI character, practical effects could've looked better" stuff Star Wars and blockbuster movies in general nowadays are known for, but the overall visual effects were very very good.
TFA = 2; AoU = 0.
- Editing: From what I remember, again, AoU had some jarring moments that seemed chopped together and took me out of the movie a bit (for example, Cap is hanging onto the back of a trailer in the South Korea action sequence, the door on which he is hanging gets shot by Ultron, and somehow Cap flies straight up into the air). I don't remember any such occurrences from TFA, so it gets this point by default.
TFA = 3; AoU = 0.
- Production Design (Costumes, Set Design, Makeup, etc.): Even though Ultron Prime and the Hulkbuster look great and the makeup and costume/prosthetic work on Vision is fantastic, and I really like the interior shots of Avengers Tower, TFA has too much variety in a good way that puts it over the top. Kylo Ren's helmet and costume design? Pretty slick. The First Order's Star Destroyer? Looks better than ever. The variety of planets from Jakku to Maz Kanata's temple/bar/hideout to Starkiller Base? Copycatted from A New Hope, but still looked fantastic and did what it was supposed to do: transport you to a galaxy far, far away. Avengers had multiple locations, but outside of the interior of Avengers Tower and floating Sokovia, none really stood out. Also, while Aaron Taylor-Johnson was plenty jacked as Quicksilver and I liked the blue bodysuit, the Nike Metcons or whatever he was wearing and his silver/platinum hair looked kinda whack.
TFA = 4; AoU = 0.
- Script (Plot and Dialogue): TFA is a re-skin of ANH, we all know that. AoU may have bogged itself down too much by setting up Phase Three (it has everything from spoken lines of dialogue to entire deleted subplots for Civil War, Black Panther, Thor: Ragnarok, and Infinity War + Endgame. Any other Phase 3 films I'm forgetting in there?), but the Hawkeye farm sequence was a good change of pace, and Whedon's quip-happy character interactions outdo whatever Force Awakens had. Also, the Avengers Tower party sequence was top-shelf MCU stuff, whereas the reveal that Kylo was Leia and Han's son was handled poorly by hologram exposition.
TFA = 4; AoU = 1.
- Acting (Character Moments and Overall Skill Shown by the Cast): Finn has one of the best character arcs in a Star Wars film to date: a no-named (literally just a number before Poe bestows upon him his nickname) stormtrooper that fights this movie's equivalent of Darth Vader. How rad is that?! Unfortunately, besides Han and Chewie coming back and Kylo being somewhat of a different Star Wars antagonist (a conflicted child vs the hulking supervillain that Vader was in ANH), none of the new characters really shine IMO. Acting chops were plenty fine though. The MCU, however, has some of the strongest and most charismatic characters ever to come across the silver screen, and their interactions together, as well as the always great big three of Robert Downey, Jr., Chris Evans, and Chris Hemsworth bringing oodles of chemistry to each scene they are in plus James Spader's voicework give AoU an easy point.
TFA = 4; AoU = 2.
- Purpose: What I mean by "Purpose" is this: did each film set out to accomplish it's cinematic universe's objectives, and how well did each film do that? How Star Wars and space-opera-ish is TFA, and how MCU and superhero-escapist is AoU? TFA basically recreating the original Star Wars, taking the audience to multiple worlds and a giant satellite that kills planets, all while shooting lasers and swinging lightsabers is as Star Wars a film as one can make. With AoU, Whedon tries to catch lightning in a bottle like he did with Avengers 1 a second time; Joss succeeds the most during the quieter, smaller moments when the characters are verbally clashing back-and-forth, and the set-pieces (especially the Hulk vs Hulkbuster sequence) are top-tier action blockbuster filming, but Whedon had too much on his plate between the studio-mandated plot and world-building to the phase 3 solo films and awkward pacing to the final Sokovia climax. Phase 1 was a like a series of bank deposits all building up to a big withdrawal in Avengers 1, whereas Phase 2 was the opposite: solid solo films for the most part that should've equaled another big spectacle film in AoU, especially after Winter Soldier and GotG Vol. 1. Plus, TFA holds the box-office record for domestic gross and completely renewed interest in Star Wars. Again, AoU was a disappointment compared to both Avengers 1 and its Phase 2 film counterparts.
TFA = 5; AoU = 2.
- Personal Enjoyment: Last brief (haha, I know) point that I have is my personal satisfaction with the film. Though AoU was a disappointment to me, I enjoy more of the superhero action/fantasy subgenre formula that the MCU delivers than the current Star Wars sci-fi space opera flavor of film. And certain images and line delivery like Thor choking Tony Stark, Vision wielding Mjolnir, and James Spader spewing out Ultron's want to end humanity in seemingly Biblical proportions ("Ask Noah."

with a nice soothing baritone put AoU over TFA for me personally. Rewatch ability for both movies is pretty low though.
TFA = 5; AoU = 3.