The concept of marketing is the overall idea of marketing and how to go about marketing well to customers. It's a five step plan that is broken out as an overarching idea into the concepts of production, selling, marketing, product, and societal marketing. These are all fundamental aspects of the overall concept of marketing and that holistic idea is different than "the marketing concept," which is only one facet of the overall equation.
The marketing concept is the idea that a business should tackle the task of meeting customer needs by producing a product that helps their lives, is useful, and/or is better and more effective than competitors' products.
The marketing concept is generally what people think of as "marketing" because it involves brand positioning, advertising, and similar actions. The concept of marketing is the holistic concept that encompasses every aspect of marketing, from product creation, to the method of sales, to the societal impact, and to word-of-mouth advertising.