We were asking for seats at a restaurant in Kyoto and were unsure whether to ask for nimei or futari. Which would be more appropriate? What are the distinctions in usage between these two counters?
Short Answer: If you're a customer trying to make a reservation, you can use whichever you want.
Long Answer: 名 and 人 are different in formality. But 名 and 人 are not keigo; they are not honorific/humble expressions. Let's not mix them up.
名 is clearly formal and businesslike as compared to 人. If you used 名 in casual conversations, you would end up sounding fairly awkward. On the other hand, 人 is neutral as far as formality is concerned. Basically it can be used in any situation.
名 is not particularly more respectful than 人. Using 名 does not automatically mean you are paying respect to the people counted. In other words, you can safely use 名 to refer to yourselves. It's not rude at all.
If you're a customer, you don't necessarily have to use 名, but it may be safer to do so if you're unsure. The more expensive and high-class the restaurant is, the more you may feel like using 名. I would probably use 人 at a cheap and casual izakaya. The difference is small, anyway, so you don't have to worry too much.
If you're an employee of a restaurant/hotel and want to count the number of guests, you usually use the formal 名, but that's not enough. 名 must be followed by 様 to show your respect to your guests (e.g., 2名様). You can also use the almighty 人 and say お2人様 (ofutari-sama), which is respectful enough.
People seem to think 名 as a quantifier is a polite and formal way of saying of a quantifier 人. However most Japanese people would rarely mind the little difference. So I think you may think that there is no difference between them. You may say the word as you like.
This is essentially the difference between using polite language and informal language.
If the staff asked you how many people are in your group, they would say something like 「何名様」 because (from their point of view) you are the customer and you should be addressed in a very respectful way. On the other hand, when you answer you should use humble language. So using 「２人」 is correct.