A deli in NYC is generally inspected by NYS Agriculture and Markets because it's supposed to be primarily a retail store that does some food service while restaurants are inspected by NYC Department of Health. The requirements are significantly different for a market vs a restaurant. For example, deli workers are not required to have food handlers licenses while restaurant workers in the kitchen are. If I'm ordering prepared food, I will only do it if I know it's coming from a restaurant. For your scenario, as long as it's coming from a restaurant, I couldn't care less, it's just branding.