Both codes serve a similar purpose but are used by different logistics companies.
EAN stands for "European Article Number" and replaces the former Global Trade Item Number (GTIN). EAN codes are recognised throughout Europe and can be used in retail, supermarkets and other sales areas to identify and assign a product within a merchandise management system. They are assigned only once and can be used by any company that sells this product.
FNSKU codes, on the other hand, are Amazon's own internal warning and label numbers and a combination of ASIN and seller identification. On top of the unique allocation and identification of products, they are used to identify the respective seller of any Amazon stock.