I just realized that I referenced my desire to go to school A in a letter to school B (in fact, I did this by mistake for 2 schools!). Most unfortunately, I have already submitted the applications for these 2 schools so I can no longer change the letters. How damaging can this mistake be? The annoying thing is, I mention specific professors I want to work with from both schools, and these names are correct, but it's just the school name that was mixed up in the editing stages.
As a long time graduate program director, I've read thousands of these essays; this mistake happens in maybe 1 in 50 or so. I have already seen one this year, for example.
Here's the deal: If it's a generic statement of purpose that is not tailored to the school in question, yeah, it hurts a bit. It's an added signal that you're not all that serious about this particular school. If your statement of purpose is well tailored to the program -- or if you've created other contacts there, e.g., talked extensively with a faculty member interested in recruiting you -- the harm should be minimal.
To add to the existing answers, I think it is a good idea to find out who the graduate director is for School B, send him/her a very short e-mail apologizing for the mistake, and note that you really are interested in School B as is hopefully clear from the rest of your letter.
These applications go through various layers of bureaucracy, but (at least in the cases I'm familiar with) the decision-making is done my the department in question. If your e-mail doesn't ask for any commitments from them, and doesn't demonstrate that you haven't read their application instructions, then (in my opinion) it is more likely to be helpful than annoying.
It also depends on the pool of applicants. If it is a competitive program with many applications, trivial mistakes can carry more weight. A redeeming quality is that the facility's names are correct. Hopefully the committee will keep one eye closed.
Edit: sad to see downvotes without comments. Disliking my answers is absolutely fine but I welcome and would love to know your thoughts. This is, after all, an academic board.
I myself have been serving in different admission committees for more than 3 years and I have to say, as naive as another answer pointed out, rejecting an application for writing the wrong school name has been one of the contributing reasons. It's never the sole reason, but it's definitely a negative trait.
Not an uncommon mistake. Good for a little laugh. Harmless.
Edit: I've read some of the comments to my post as well as other people's responses. If it were indeed true that someone might hold this against the student, I would find it naïve at best. Do we not know that students apply to multiple schools (if they know better)? Do we expect some sort of early allegiance from them? Do we measure a student's level of commitment, never mind his/her research abilities, by a small oversight buried in the SoP? None of this is very sensible.
The OP clearly states that he tailored his SoP to each school, so he did his due diligence and as a result the SoP is presumably informative as to the candidate's fit. No matter how you cut it, the application will (should) be accepted or rejected based on the qualifications of the student and the strength of the application package, this for the sole benefit of the graduate program.
The only case where I could even conceive this being harmful is if the package were marginal (for the school's standards) to begin with. I can't remember a single case (out of thousands) where I made a recommendation/decision based on something like this.
Read Corvus's careful response: Harm ranges from "a bit" to "minimal." I'd argue that, when the chips are down, it is very close to zero or just zero.
The statement of purpose is very important, but if your CV is rock-solid, with great undergraduate scores and strong references, the importance of your SoP gets a bit lowered. So, yes, what happened is embarrassing, and can reflect negatively on you, but don't lose hope. And it's probably more common than you think.