The pirate (from Greek "peiratès", which means "the one who undertakes", "the one who tempts fortune") acts for his own account, he is an outlaw who cross the seas and who plunders, violates and often kills without distinction of nationality. If he is caught, he is hanged high and short. High so that everybody sees him, and short to save some rope!
The privateer acts on a letter of mark delivered in the name of the king (often, he wasn't even warned about it). This letter is a document by which a country recognizes him as an auxiliary military force. The privateer acts for the service of his country. If he is captured, he shows his letter of mark, which avoids him the rope. But some little scrupulous privateers took advantage of this official paper to plunder and kill the traders, like the pirates. The piracy is as old as the hills and still exists, whereas privateers raged only during three centuries (from the XVI to the XIX century).
The freebooter (or "Brother of the Coast", from Dutch "vrijbuiter"; which means literally "free booty maker") is a privateer of the Antilles which attacked the Spaniard in the XVII and XVIII century. The word appears at the end of the XVII century (Furetière, on 1690), but it is usual since Dutertre in 1667.
The buccaneer (derived of the Caribbean "boucan") was originally a hunter of wild animals. He treats the meat by a smoking process called "boucanage" learnt by the Indians Arawak and he trades animal skins. At first, the buccaneer occupied the lands on Hispaniola island, which belongs to Spain (today Haiti and the Dominican Republic). The rarefaction of the game on the islands as well as the Spanish's attempt to eliminate the buccaneers makes them join the Brother of the Coast.
The Smuggler is engaged in the clandestine business of prohibited goods or for which he did not settled the customs duties.
With many thanks to Olivier Rolko for his english translation help from https://www.pirates-corsaires.com/definitions.htm