The replacement of the lateen by the gaff-and-boom-rigged spanker (or "driver") was a long process, with different ships carrying different rigs at the same time, but there's no doubt that it happened because the gaff-and-boom rig was more efficient.
The big drawback to the lateen rig (please note the spelling - no N in the first syllable) was, as I said in my last post, that the sail had to be furled and reset every time the ship came about. (The process also necessitated casting loose and re-rigging several important lines, such as the "bowlines" that controlled the forward, lower end of the yard.) The first lateen sails seem to have been triangle-shaped, with a temporary, rectangular extension, called a bonnet, added to the bottom when the wind was light. The lateen sail actually became a quadrilateral while the yard retained its full length. In about 1680 the bonnet fell out of favor, and the bottom of the mizzen itself was extended to compensate. (My source on that point is James Lees's The Masting and Rigging of English Ships of War, 1625-1860.)
The next step in the evolutionary process seems to have been the adoption of a sort of hybrid system: the full lateen yard was retained, but the sail (still called the "mizzen") was shaped like a trapezoid, with its head secured to the yard and its forward edge to the mizzen mast. That arrangement made a big differece to the handling of the ship, in that it eliminated the necessity of resetting the sail when the ship came about. (The yard could be left on the same side of the mast all the time; which side didn't make much difference, though contemporary models usually seem to show it on the starboard side.) Mr. Lees says that configuration came into use in 1730, and remained common in smaller ships until 1745 and in larger ones till 1780. (Those dates, as Mr. Lees undoubtedly would be the first to agree, are approximate; he's talking strictly about English warships, and in any case there were no universal rules about such things.)
The boom-rigged driver (or spanker) seems to have been introduced originally as a fine-weather replacement for the lateen-rigged mizzen. Falconer's Universal Dictionary of the Marine, 1780 edition, defines the driver as "an oblong sail, occasionally hoisted to the mizzen peak, when the wind is very fair. The lower corners of it are extended by a boom, or pole, which is thrust out across the ship, and projects over the lee quarter." The head of the driver often was extended by a short "driver yard," rigged (rather like a studdingsail yard) with a halyard running to the peak of the mizzen yard. (There's a good drawing of that configuration in Mr. Lees's book, p. 112.)
Contemporary references from the second half of the eighteenth century can be confusing. I bumped into this particular problem when I was working on my model of the frigate Hancock, which was launched in 1776. A contemporary description of her by Commodore Sir George Collier, in a letter to Admiral Lord Howe after her capture by the British, says she "has a mast in lieu of an ensign staff with a lateen sail on it has a fore and aft driver boom with another across two top gallant royal masts pole mizzen topmast a whole mizzen yard and mounts 32 guns...." There's room for some interpretation there (partly due to Sir George's disdain of punctuation), but it's pretty clear that she had both a full lateen mizzen yard and a boom-rigged spanker - as well as a small lateen sail set on her ensign staff. When I was working on that little model I talked those contemporary references over with several other folks, who had just as much trouble sorting them out as I did. (We never did figure out exactly what that phrase "with another across" meant.) Our eventual conclusion was that the Hancock had a total of three sails aft of her mizzen mast: the mizzen (with its head laced to the lateen yard and its forward edge to the mast), the driver (with, when set, its forward edge secured to the mast, its head secured to the mizzen yard and the short "driver yard," and its foot to the boom), and a small lateen yard set to the ensign staff, sheeted to a small boom that, I think, ran out through the middle of the carved rattlesnake on the transom.
Most of those features are confirmed by the series of paintings depicting the capture of the Hancock, attributed to Francis Holman and now in the Peabody-Essex Museum. Those paintings show a total of five British and American warships. Most of them have full-length lateen yards on their mizzen masts, though the Hancock, disturbingly, appears to have a gaff - and no boom. (That sort of contradiction between written and pictorial sources, unfortunately, is pretty common. Holman probably never actually saw the Hancock; Sir George Collier did.) One of the ships in one painting has her driver set and her lateen mizzen furled.
That whole arrangement does seem to be unnecessarily complicated. The Holman paintings suggest that the lateen sail on the ensign staff actually was pretty large. Handling it when the ship came about must have been quite an exercise. The whole ensign staff, with the lateen yard mounted on it, would have to be struck, and reset when the ship settled on the new tack; otherwise the boom, as it swung across the quarterdeck, would clobber the ensign staff. Well, warships of that period had relatively big crews - but the scene on the Hancock's quarterdeck when she went about must have looked like a rugby match.
Mr. Lees gives 1810 (again, we need to take that as an approximation) as the date by which the gaff-and-boom-rigged driver (or spanker) became a permanent fixture, and the term "mizzen" ceased to be used as the name of a sail. By this time the crossjack (or crojack) yard - a "square" yard, occupying the same position on the mizzen mast as the fore yard and main yard occupied on their respective masts - had been in use for more than a hundred years, but rarely if ever had a sail set to it. (Neither the Victory nor the Constitution set a sail on her crojack yard. Its principal function was to provide a means of sheeting the mizzen topsail.) It seems, in fact, that sailing warships even in later periods rarely if ever set sails from their crojack yards - though merchantmen certainly did from the middle of the nineteenth century onward. (The Cutty Sark set a crojack, as did most of the famous American clipper ships. It's easy to see why the crojack wasn't more popular, though. When the ship was running before the wind, the crojack would partially becalm the main course; when working to windward, the spanker would becalm half of the crojack. It's fairly rare to find a photo or painting of a ship that doesn't have either the crojack or the spanker furled.)
None of this is directly relevant to a French seventeenth-century galley, which unquestionably had a straightforward, old-fashioned lateen rig (and a relatively large crew to manhandle the lower ends of the yards around the masts when the ship tacked). But there's no doubt about why the lateen rig fell out of favor in Western ships. The gaff-and-boom rig was easier to handle - and, with the addition of the mizzen staysail at about the same time the mizzen yard got chopped off, offered almost exactly the same amount of sail area. The most bewildering part of the story is why the full-length lateen mizzen yard, with no sail hanging on the part projecting forward of the mast, lasted as long as it did. At least one contemporary source (I confess I've forgotten which one) suggests that keeping the traditional mizzen yard intact was a good idea because it might come in handy as a spare main topsail yard. Sailors in those days were good at coming up with excuses for sticking with the old ways of doing things.