The internet at large has already covered a lot of what's wrong with BvS, so rather than repeat what's already been said, I'm going to restrict myself to a couple of major things not much touched upon, specifically: the Act 3 climax is a huge mis-fire; and the post-climax codicil makes no sense whatsoever.
First, the climactic battle with Doomsday. The problem here is: who really cares? Comparisons to The Avengers are inevitable, and in that movie we've had all kinds of foreshadowing and build-up: the Avengers must stop Loki from using the Cube and opening the portal, failing every step along the way, until the emotional climax of Stark laying down his life... In BvS by contrast we get a rock-monster with arbitrary powers and an equally arbitrary weakness, that appears deus ex machina (quite literally) with no motivation nor character of any kind, and that is not set up in any emotionally meaningful way by preceding events or threats. And even ignoring the disconnect from the foregoing story, there's neither a logical reason that the Kryptonian spaceship even has the ability to create a Doomsday nor a narrative reason that Lex chooses to do so (contrast Avengers, where opening a portal is Loki's motivation from the very beginning).
Consequently, here there's no sense whatsoever that we're building towards this apocalyptic battle -- a problem highlighted by the fact that Wonder Woman decides to get involved in the fight for no adequately explored reason. (More generally, Wonder Woman is woefully underdeveloped -- and not in an intriguing, "show me more backstory!" kind of way, but in a frustrating "what does she want? why is she doing that?" way.) In fact, given that Luthor's main plot is all about manipulating Superman and Batman over many months into fighting each other, the whole Doomsday plot line feels like it was left over from an entirely different draft of the script. Having made the Batman v Superman conflict the core of their movie, the writers apparently had no idea what to give them to do once they had resolved that conflict.
Second, the post-battle State funeral. Why? In the Death of Superman comic book source material, this makes perfect sense. In the comics arc, Superman is a long-established hero, known and trusted, even loved; and the world watches as he fights Doomsday all the way across the country for days on end, other heroes falling by the wayside, until finally, battered into exhaustion in full view of friends and news cameras, he sacrifices his life to save the world. Of course the world mourns. But in BvS, (i) Superman is mysterious, distrusted, and even disliked; (ii) Doomsday appears out of nowhere and spends around twenty minutes in Metropolis, hardly enough time for everybody to decide that we've tried everything and the world is going to end unless Superman can stop it (frankly, anybody that was there for Zod is probably thinking "meh, I've seen worse"); and (iii) nobody witnesses Superman's self-sacrifice and death except Batman, Wonder Woman, and Lois Lane... but their word is good enough for the US government to throw a funeral fit for a president.
On reflection, the two best sequences in Dawn of Justice are (i) Batman rescuing Martha Kent, and (ii) Wonder Woman fighting Doomsday. The former is the one fight scene that is most true to the Batman character (I wouldn't be at all surprised to learn it was done entirely by the Second Unit); and the latter the only part of the movie where anybody seems to be having fun. Like Doomsday, Wonder Woman seems to have wandered in from the theater next door where she had been starring in a movie that was a lot more fun than the one I was sitting through.
And so it struck me: DC could in fact have made a far more interesting movie if Superman never appeared at all. Sure, he's out there in the world somewhere, motivating Lex and the others to their actions, but never actually seen. Edit out every scene with Kent or Superman (except maybe Bruce's nightmare sequences), give Wonder Woman some proper background and motivation, and you've probably got a pretty decent movie about how the rest of the world feels about Superman, and how it copes when he doesn't come flying to the rescue.