Often a product refers to an object that you can find on a store shelf, or some thing that is sold. Often a production refers to the result of the work of a large team in putting something together, for example a show. Production also refers to the process of producing things. For example you can talk about production rate or efficiency of production, or a production plant where the production of products occurs. There may be also other meanings, but this is what comes to my mind. I hope it helps : )

product 是产品 production 是产制

Product is also occasionally used to refer to something or someone who is a certain way because of some cause, usually in a negative sense. For example: This depressed person is the product of a malfunctioning society. The rise in crime is a product of the new easier immigration policies.