The expiration date in a quote is used to determine how long the quote will be kept live for. E.g. if you wish for the customer to only be quoted at this price until a fixed date, you would set this date in the 'Expiration Date' field.
Date required defaults to next day and can be changed accordingly if you wish to keep the quote for a fixed period before turning it into an order to be purchased (in which the date required is carried over to purchasing)