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Vessel is a catch-all term, like 'watercraft', which describes any floating object used for the carriage of people or goods. Generally smaller and less complex vessels are 'boats', whilst larger and more complex vessels are 'ships'. As a general rule, you can put a boat on a ship, but you can't put a ship on a boat.
I will not attempt to explain at any great length what the difference between a boat and a ship. For the sake of simplicity, all the sailing vessels discussed here are ships. And we will say a boat is a small, open, nautical vessel used to get from ship to ship, or ship to shore, or for fishing with grandpa..