Ooh, that's a little subtle, and I never really thought about it seriously. If there is anybody on this forum who is in Marketing, they could give a good answer. But from my point of view ...
I think "make" is the manufacturer, and brand is the name we know it by. So Renault would be the make of a car, but Twingo would be the brand. Canon would be the make of a camera, PowerShot would be the brand.
But in everyday conversation, people might be sloppy about this usage.
BTW, this thread really belongs in the "English only" forum.
A "brand" is a name used by a manufacturer of any product. Franco-American is a brand of canned pasta (it happens to be one of several brands Campbell's uses to sell its products).
"Make" is used almost exclusively for cars, and usually in conjunction with the word "model." Ford Taurus, Mitsubishi Eclipse, Toyota Celica--the "make" is the first word in the pair, the "model" the second word. Just as a single company can produce products under several "brands," an automobile company can produce many makes of cars; for example, Ford, Mercury, and Lincoln are all makes produced by Ford Motor Company; General Motors produces Pontiacs, Chevrolets, Saturns, GMCs, among many others.
You would never ask a person "What brand of car do you drive?" You would ask, "What make of car do you drive?" For most other products, you would never ask for them by "make," you would ask for them by "brand."
You are quite right. There are two ways of using the word "brand".
In common usage, it is the name of a product. "What brand of cigarette do you smoke?" "What brand of hot dog do you want me to buy?" "What brand of paint did you use on the house?". But you are right, you would not use brand for a car.
Then there is the marketing term. Marketing people use it to mean establishing a unique identity to an item. First and foremost is the name. For marketing people, and "marketing talk" is now very commonly heard, the model name of a car is the brand, just as the name of a hot dog is the brand.
Make is a specific thing usually and refers to a physical type of object often something mechanical - brand is more ethemoral E.g. The Brand of Coca Cola not the Make of Coca Cola although Coca Cola may make Coca Cola - the make of car made by the BMW brand or the make of spanner or dishwasher-