According to the research discussed in Daniel Kahneman's Thinking, Fast And Slow, our cognition is geared towards stereotyping. The main example from the literature was a description of a woman who was given a short profile, and participants were asked to rate how likely each statement was. It was highly likely that she would be active in the feminist movement, not likely that she was a bank teller, but more likely that she was a bank teller who was active in the feminist movement.
The fallacy is that there's a smaller set of bank tellers who are active in the feminist movement than there are bank tellers in total. So it stands to reason that it's more likely that she is a bank teller than both a bank teller and being active in the feminist movement. The mistake is that although there are less feminist bank tellers than there are feminists, being a feminist fits better with the profile than bank teller does.
This cognitive tendency seems to me a good explanation for a persistent apologetic - the cry that whenever anything bad happens under the name of religion of "that's not religion". (Or the slightly more sophisticated version - "that's not true religion".)
This apologetic is hardly limited to anti-atheist apologetics, but has served a useful purpose throughout history of branding other religious people who disagree as atheists. I used to think of this move as nothing more than a rhetorical ploy, but now I've come to accept that it's (mostly) uttered in full sincerity.
The fallacy was made explicit by Antony Flew with the No True Scotsman fallacy, but making the fallacy is not the same as the reasons why the fallacy is made. And because it's an informal fallacy, it might not be at all obvious why it's fallacious to distinguish between the two cases. Is the critic arguing engaging in equivocation, or is there some special pleading going on by the proponent? A deeper examination is almost always needed.
One of the things that's prominent in the critics of religion when faced with the apologetic is to ask what the difference is. This seems the right question to ask. While it might just be obvious to someone that two concepts don't share a resemblance, illustrating that difference is another matter.
The importance of being able to correctly identify where the criticism lies is the only way to make sense of the criticism. Even if the criticism is misguided, it's important to identify where the thinking has led astray - there's no reason to think that critics are any less biased in this respect than proponents. The important point is to recognise the need for rational exploration because our biases, make it all too easy to shrug off what seems like misguided criticism.