As far as I understand, while accepting offers, sellers will have the the fair price on their mind and may sell to the person who first offers that price. Seller has the right not to sell the item if he or she isn't satisfied with the price for it.
In the other hands, in the auction, the buyers/ customers will decide the price for the item, not the seller. The price will depend on the demand on the item. Seller does have to sell the item even if the price is not good.
Each way has its own advantages and disadvantages. I think some "making offers" sale out there are turning into auction while the sellers still have the right to cancel the sale
