It depends on what you want the answers to be like. There's a certain difference in tone between MO and MSE posts stemming from the expected audience. MO is full of real deal, bona-fide, research level mathematicians; MSE has some of those, but also undergraduates, math hobbyists, and research types from other fields. Answers on MO assume a certain level of background knowledge (or rather mathematical maturity) that isn't implicit here. (Another way of saying this would be, to borrow some terminology from one of Terry Tao's essays, that MSE answers are written for pre-rigorous and rigorous audiences, while MO is aimed towards the post-rigorous stage.)

As for your example, I think an etale cohomology question would actually be fine on either site. On MO, I would expect the answers to be longer, less detailed (in terms of formulas and definitions), and more reflective on relationships to other areas of math. An etale cohomology soft question on MSE would more likely be focused on getting the main ideas across. I'd expect to see more analogies to lower level math, more references for "if you want to learn more about the subject," and in general a tone that assumes the question is being asked by a student, rather than a mathematician rediscovering his own field.

It depends on what you want the answers to be like. There's a certain difference in tone between MO and MSE posts stemming from the expected audience. MO is full of real deal, bona-fide, research level mathematicians; MSE has some of those, but also undergraduates, math hobbyists, and research types from other fields. Answers on MO assume a certain level of background knowledge (or rather mathematical maturity) that isn't implicit here. Another way of saying this might be, to borrow some terminology from one of Terry Tao's essays, that MSE answers are written for pre-rigorous and rigorous audiences, while MO is aimed towards the post-rigorous stage.)

As for your example, I think an etale cohomology question would actually be fine on either site. On MO, I would expect the answers to be longer, less detailed (in terms of formulas and definitions), and more reflective on relationships to other areas of math. An etale cohomology soft question on MSE would more likely be focused on getting the main ideas across. I'd expect to see more analogies to lower level math, more references for "if you want to learn more about the subject," and in general a tone that assumes the question is being asked by a student, rather than a mathematician rediscovering his own field.