All rooms on the club floors, as well as some rooms elsewhere (such as suites), get access to the Regency Club (RC). As it happens, the GH Shanghai has one of the better RCs. A decent breakfast is served every morning there, and snacks sufficient to make a light meal, along with drinks, are served in the early evening. Plus the club has an excellent view of the Bund across the river. Amenities are better in club rooms as well. Without doubt, if the price difference was reasonable, I would pick a club room.

Welcome to the Hyatt forum, btw. There is an excellent review thread here on this hotel and I am going to suggest to the mods that this thread be made part of it.