Customer research and market research could be both different. Market research usually involves researching about market conditions. This may also involve market studies such as supply of the product and its relationship with the demand of the product.
Well, I believe that the most important thing is that both of the research is something that is going to provide you with the information that you are going to use to take advantage of the market in making sure that your business remains profitable by getting the best from your customers and staying ahead of your competitors.