I write this as an inveterate nerd who on the whole actually really liked Star Trek Into Darkness.
But Fast & Furious 6 is better. Much better. I know, I know. But hear me out.
1) With Fast & Furious 6, what you see is what you get. I'm not going to try to convince you to see it if you have no interest in popcorn-y action movies. The trailer makes the movie look ridiculous for the pure and simple reason that it is. There's a character who used to work for the good guys but now works for the bad guys because she bumped her head and lost her memory. The climax of the movie features cars dangling from a plane, for Chrissakes.
But it works. The movie promises fun, car chases, and explosions, and that's what it delivers. Then look at Star Trek Into Darkness, which—at least in the marketing—highlighted as a major draw of the movie Benedict Cumberbatch's villain. He's such an interesting character, we heard time and again. (A super-smart terrorist who uses overly elaborate plans to stay one step ahead of the good guys? How many times have we seen that in the last year alone?) There's a big twist relating to his true identity! And, yeah, rebooting one of the franchise's most iconic villains is a big deal. It's important. It's epic. It's exciting.
But the way it was handled in the movie was just kind of... blah. He wasn't a bad villain. But the great parts of Star Trek Into Darkness, for me, were the parts that explored Kirk and Spock's character development, and for all that Khan was presented in the marketing and the movie as OMG The Scariest Villain Starfleet Has Ever Faced This Is So Epic and Alarming, I didn't buy it. The part of the movie I enjoyed the best was the third act, after Khan had been defeated (but before he died) and all the changes that had happened to Our Heroes over the past hour-ish finally came to a head.
Star Trek Into Darkness brought a pound of the highest-quality ground beef, and all they could give us in return was a serviceable cheeseburger. But with Fast & Furious 6 we wanted a nice, juicy cheeseburger (emphasis on the cheese), and that's what the writers gave us.
2) Related to point one, all of the elements of Fast & Furious 6 mesh together really well, while with Star Trek Into Darkness... not so much. It's a movie about car chases, but it's also a heist movie, a movie wherein people beat the shit out of each other (former MMA champ Gina Carano is in it, of course it is), and a movie about the power of family. That last one serves as the emotional core of the movie, but aside from popping up in a few scenes it generally keept itself belowground. This is a Fast & Furious movie, after all. People don't pay the ticket price to see drama and romance. There is a romantic element, but the lady in question is Michelle Rodriguez, so you can be sure its sappiness level (close to zero) fits the movie as a whole.
Likewise, the heist element drives the plot along, but at the climax it turns out that—surprise!—the villains' plot means the good guys and bad guys alike will have to engage in a high-speed tank/motorcycle/car chase both on a highway and on an improbably long runway. Is it ridiculous? Fuck yeah. But it works in the context of the movie. I never batted an eye.
There were elements of Star Trek Into Darkness that I didn't like (see: KHAAAAN!), and there were parts that I did (SPOOOOOOCK!), and while the latter outweighs the former I still don't feel that they worked well together. I just wanted to watch Spock and Kirk be conflicted and Scotty be competent and Uhura be better than anyone and Khan and Admiral Marcus kept waving their hands in my faces going "Look at meeeeee! Look how cool I am!!!"
3) Speaking of Khan, I'm not going to get into the whole whitewashing issue—Zoe did that better than I could in her review on The Mary Sue—but I do want to talk about diversity in general, because Fast & Furious 6 has it. There are six or so main good guys, and most of them aren't white. It's not presented as a big deal, it's just that they're part of an international heist team with members that originally come from all over the world so, y'know. Not all white dudes.
You can make the argument that there's some stereotyping involved, particularly with the character played by Tyrese Gibson. This movie is not some great statement on race relations or tolerance or anything like that. But at least number of non-white actors in the main cast isn't a grand total of two.
4) Fast and Furious 6 has kickass, sexy ladies who actually do things. As with the diversity issue, Fast & Furious 6 isn't some progressive portrayal of gender. Quite often women, particularly extras (looking at you, pre-car race party scene) are presented as eye candy. But it's better than Star Trek Into Dakrness, not least because Fast & Furious 6 just has more female characters. (You'd think that a franchise that celebrates diversity would have more than just Uhura and Carol, but hey, what do I know?)
More than just being an issues of numbers, though, the women in Fast & Furious 6 take control of their own plotlines. Hell, even when Giselle (Gal Gadot) and Riley (Carano) use their ~feminine wiles~ to get information on the baddies out of a car mechanic, they're the ones who chose to do that. Is the whole "har har, women use sex to get things and men think with their dicks" thing dumb? Yes. But it's an action movie.
The female characters also also actively engaged in the plot at a level that Carol and Uhura in Star Trek Into Darkness really aren't. Giselle, Riley, and Letty (Rodriguez) get into fights, they shoot guns, they race cars. Everything the men do, pretty much. (There's a fun scene where Riley and Letty are kicking each others' asses at the same time two of the male protagonists are getting theirs thoroughly beat by a lone baddie.) Carol Marcus is introduced as an advanced weapons whiz and then stands around for most of the movie, never really getting to show off her intellectual chops (her underwear is another matter). And Uhura, who's always been excellent, got the raw end of the stick; in a movie filled with character development for its protagonist, for most of the movie her role was that of "Spock's girlfriend." She got a chance to shine on the Klingon home world, but unless I'm forgetting something all the other substantial screen time she got was Spock-related.
Conclusion: If you're in the mood for something fun, exciting, and summer Hollywood blockbuster-y, I'd recommend Fast & Furious 6 over a repeat viewing of Star Trek Into Darkness. You don't even need to see any of the previous movies (even though Fast Five, the only other one I've seen, was good. The Rock says "funderwear"). Go on, give it a chance.
(Thanks to @HumbugJones, who commented "I liked it more than Star Trek" and set my mind a'runnin'.)