Face value can be also called par value or nominal value, which is the value that we have given to anything based on what is printed on it or based on what is the initial declared value of it at the time of creation. If we take the examples of some items that can be related to monetary value, for example, if we take a coin, a currency note, a stamp paper or a bond paper, face value is the legal value. It is the price given for that item or the price printed on that item at the time of creation or at the time of presenting it to the market.
Once an item is presented in the market with a face value, the market starts increasing or decreasing the price of the item based on its expected future value, supply and demand and based on the utility that it provides or based on some other factors. The change in price which is given by the market against the initial face value is called market value. Hence, the face value and market value can be totally different.
You can find some rare coins or notes having a legal value at the time of creation trading in the market with a totally different higher value. Sometimes a note that is having a face value can also trade at a market value below its face value.
Coins will also have salvage value because of the metal used to create that coin. The legal value of the coin can be less but due to the higher value of the metal used to create that coin, the market value will be higher than the face value or legal value of that coin.
So in simple terms, face value is the initial basic price and market value is the present price at which an item is traded in the market.