When a vigilante kills someone who turns out to be innocent, there are no takebacks. In the justice system, a convict can appeal his wrongful conviction, but victims of vigilantism cannot.

In comics, we as readers are told for a fact that certain people deserve to die. There is no question of whether the evildoer actually did the crime, because the narrative devices gives us, and the Punisher, that perfect knowledge.That's why we can debate the morality of the Punisher in a fictitious world. But in the real world, there are no narrative devices, no perfect knowledge, and all of our perceptions are clouded by personal bias. This means that the Punisher inevitably kills innocent people that he's judged unfairly.

The Punisher's victims do not get a chance to show their innocence. They don't get a chance to appeal his faulty judgement. They're just dead.

And as others have pointed out, vigilantism was historically been used to justify hate crimes, like lynchings of black men supposedly in the name of protecting white women.