In a recent post on respect, Jean Kazez conflates respect and esteem: "Basically respect is esteem—it’s having a set of positive thoughts about someone or something." But I don't think that this is standard usage, although the two perhaps overlap. To esteem something is to hold it in high regard, but to respect something need not involve that. Thus, although some people treat 'self-respect' and 'self-esteem' as synonyms, the (very common) insistence on the difference between self-esteem and self-respect is not incoherent.
Consider the following scenario (based on a real-life one). A business, not satisfied with its current distributor, switches to another, and finds that they dislike the new distributor even more. "You just don't respect the barely competent as you should until you come across the truly incompetent," says one employee to another. Now, it makes no sense whatsoever to suggest that the employee means that we should esteem the barely competent, as if it were of high value; rather, he is suggesting that competence should be respected in even those forms that we wouldn't normally esteem. Nobody esteems the barely competent, for precisely the reason that it is the bare minimum of competence; but it still makes sense to argue that it should be respected, because, barely or no, it is competence.
Further we can respect things we regard negatively. Many people don't have positive thoughts about rattlesnakes, but the standard advice to respect rattlesnakes still makes sense. The advice does not suggest that they should think positively about rattlesnakes, which would be useless advice; rather, it means that they should respect them, i.e., not treat them with contempt, nor simply ignore them, and act toward them in a way that shows appreciation for the ways in which they can be dangerous.
This is perhaps related to Darwall's suggested distinction between recognition respect and appraisal respect: to exercise recognition respect is to treat something as sufficiently weighty to take into account in your deliberations, while to exercise appraisal respect is to treat something as having some sort of merit based on some particular features. (We could, if we wanted, distinguish both from esteem by suggesting that the latter is a general estimation based on all the known features of a thing.) But even appraisal respect is a bit tricky: John could respect Tom for his singing voice and, due to envy and resentment, hate it as well, regarding it as a terrible waste, or as something perverted from its proper use. This is not straightforwardly a positive assessment.
Consider the notion of 'respect for persons'. When someone says that it is a general obligation to respect other human beings, this does not obviously mean, as Kazek's reduction would suggest it should, that we should go about thinking positive thoughts about them. Telling you to respect your parents, or your children, or your friends, or your benefactors, is not telling you what to think about them; it is telling you to engage in one form of just behavior toward them.
In any case, while the question of the nature of respect is itself an important one, the context in which it arises is not. I am amused by the hypocrisy of most of this atheist indignation about the legitimacy of disrespecting beliefs; in most cases it obviously arises not because they have any interest in fulfilling obligations or counsels of respect about beliefs but because they are looking for excuses to excuse inexcusable disrespect to the believers of those beliefs. When you clear out all of these cases, leaving only the honest discussants, atheist or otherwise, in the mix, it turns out that there is nothing particularly religious about the issue; indeed, there is nothing about it that particularly pertains to beliefs. Rather, it is about what justice towards other people, even those with different beliefs than one's own, requires. That's the interesting question.