It's got plenty wrong with it. For one thing, its portrayal of antimatter is unspeakably stupid. It claims that matter and antimatter only annihilate each other, not if identical particles and antiparticles come into contact, but if identical people
do -- so matter Kirk can interact just fine with an antimatter universe, and antimatter Lazarus can exist readily in ours, but if the two Lazaruses meet, it's kaboom. Also it makes the equally idiotic claim that if a matter-antimatter explosion occurs, it will destroy both entire universes.
Which not only contradicts real physics, but everything else in Trek as well. It had already been established in "The Naked Time" that matter-antimatter annihilation was what propelled the Enterprise
itself. So this completely fanciful take on antimatter just didn't fit with existing continuity. Not to mention its portrayal of dilithium. Previous and later episodes treated dilithium as a means of focusing and channeling the energy of the engines, but TAF claimed that the crystals were themselves the power source.
The way Kirk and Spock reason out the annihilation threat is also completely inane. The leap from "matter and antimatter... cancel each other out violently" to "absolute annihilation... of everything that exists" is arbitrary and inexplicable. The cosmic-level threat is ill-justified and ridiculous, tacked on to what's really a much smaller story in a crude effort to manufacture a sense of peril. (Note that this is actually the only TOS episode or film in which the entire universe is said to be in danger, and one of the very few such instances in all of canonical Trek -- although universal destruction is a far more common threat in Trek novels for some reason.) Not to mention Spock's stupid line about how the sensors are "designed to locate and identify any object in our universe," so anything they can't detect must be from another universe. That's just so totally idiotic. How can sensors be designed to detect something that isn't yet known? I think that even as a kid I was troubled by the illogic of that line. (And again there's a contradiction of previous episodes, since the galactic barrier in the second pilot didn't register on sensors either.)
The episode also contradicts its own internal logic as well as the rest of the franchise. At some points, the switch from one Lazarus to another can only happen at the portal site on the planet surface and causes a "winking out" effect that's allegedly felt universe-wide, but at other times it happens aboard the Enterprise
with nobody noticing. And maybe there's a scene that was cut, but the first time we hear about the two Lazaruses switching is when McCoy mentions his bandage disappearing, whereupon they promptly switch back, so it's awkwardly structured.
And there are just too many ideas that we're told about but are never followed up on. Like Lazarus being a time traveller -- that never has any relevance to the story. The backstory about planetary destruction and the epic vendetta is too dependent on telling rather than showing. "Let That Be Your Last Battlefield" had a similar premise, but there we could actually see the two enemies confronting each other directly, so it was sold much better -- and if "Last Battlefield" does something better than another episode, that other episode is in trouble. It doesn't help that the mad Lazarus is just completely over-the-top and unsympathetic, whereas we see too little of the good one to identify with him. (Maybe he would've come across better if they hadn't written out his romantic subplot with Lt. Masters.)
And then there's the beard. Let's not even talk about the beard.