What distinguishes a global city?
A global city makes new norms. And two requirements for that happening are complexity and diversity. Quite often, in countries around the world, it's the most global city, especially New York, where new national and international norms are made.
Is a global city always a megacity, and vice versa?
I'm so glad you asked. Most global cities are really not megacities. Some are, but the question of size is a tricky one. Size is important for a global city because you need enormous diversity in very specialized sectors, a whole range of them. Some of the leading global cities are very large, like Tokyo or Shanghai. On the other hand, you have cities that are simply very large, like Mumbai or Sao Paulo. I don't think Lagos is a global city; it's just a huge city. You have a lot of very large cities that are not necessarily global cities.
Sassen goes on to describe the transformation of Miami from "a dreadful little spot" (her term) before the 1990s to a major global city--albeit as much a part of Latin America as of North America. Miami is now #33 on the Foreign Policy list of global cities, which is led by New York, London, Tokyo, Paris, and Hong Kong. Admittedly, there are not any surprises on that top part of the list.