Why don't respiratory doctors deal with obstructive sleep apnea instead neurologists?

They do: Usually the pulmonologist and neurologist will interpret the sleep study and the order CPAP treatment.

Both: In my practice, i get referrals from both neurologists and pulmonary specialists. I tend to especially get the referrals of patients who cannot tolerate the CPAP machine or are not severe enough to warrant a cpap. Both of these specialties are invaluable!

CPAP: As a psychologist, i've heard from patients that there are many cognitive benefits from cpap. Improved memory and attention are just two. Of course, these might improve because the person is not as tired anymore, but i'm willing to bet empirical evidence will show cognitive enhancements.

Both: Both specialists evaluate and treat OSA patients. Often depends on community referral patterns.