I agree with quochuy0706 on his definition of customer. But I would say that a consumer is anyone who consumes goods or services (which is basically everyone). But I've seen the term consumers used in marketing to refer to specific subgroup of people that consume a particular good or service.

For me, I don't think it have a big different, on the other words, they have the same meaning, customer is a person or an organization that buys something from a shop/store or business and consumer is a person who buys goods or uses services :)