Before the Seventh Umbral Era, the power structure in Limsa Lominsa was a lot simpler. Much like today, there were captains, and whoever won the Trident (big race, few rules) won the Admiral's chair. Much like today, the Admiral was essentially a regular captain, but with more legitimacy because they commanded Lominsa's affairs, not just their own. And, much like today, the Thalassocratic Navy was essentially the crew of the city itself, loyal to the people and the current admiral, not a specific captain.
However, that was the top of the food chain back then. Not only that, but the admiral often filled the highest ranks of the Navy with his own men, anyway, so some people didn't see much of a difference between them and the pirates - one just wore a different uniform and also got to command Limsa's resources and had to manage Limsa's concerns. Other crews, especially, grew disdainful of the "legitimate" authority once its highest ranks swelled with the new Admiral's old hands and thus took on their idiosyncrasies to some degree. To really get an idea of how we get from there to where we are now with the Yellowjackets, though, is going to require a bit of a digression.
The Thalassocratic Navy's membership was known as The Knights of the Barracuda. As a thalassocracy, Limsa's power is rooted in the power over the seas, and thus the Navy sails all the Five Seas of Eorzea. There's a squadron for the Bloodbrine Sea (2nd), the Indigo Deep (4th), the Sea of Ash (5th), and the Sea of Jade (9th). Lominsa's sea, the Rhotano, is subdivided with a squadron for every major area of their region. There's a squadron for Galadion Bay (1st), the Straight of Merlthor (3rd), the Cieldalaes (6th), the island of Bianaq (7th), and one for Clearwater (8th). You'll notice that Clearwater is an area around Coerthas and the Black Shroud, but it connects directly to the Straight of Merlthor and the Rhotano by way of the White Maden.
Each squadron is broken down into Levies, with some infantry acting as police for the area in which they are stationed. These are the Yellowjackets. For instance, 1st Squadron 2nd Levy Infantry Yellowjacket Sergeant Sylskaetsyn is a police sergeant of the 2nd Levy of the Squadron stationed in Galadion Bay's infantry. Even in 1.0, these Yellowjackets acted as the police within areas controlled by Limsa Lominsa.
Then came the Seventh Umbral Era.
When the Maelstrom was reinstated and all of Limsa Lominsa's resources were brought under one banner in hopes or surviving the Calamity, the Admiral was elevated to Chief Admiral, and the Maelstrom swallowed everything; the Thalassocratic Navy became only a division of her authority. Maelstrom Command oversees the Thalassocratic Navy ('Cudas), but also The Crimson Fleet, the city's principle fleet independent of the Navy proper, the Privateer factions (civilian ships commissioned or requisitioned for the war effort). Maelstrom Command even controls the pirates, to some degree, as part of the Galadion Accord (an agreement that united the pirates as one under the pirate lord Hyllfyr, who has agreed to fight under the Crimson Banner of the Maelstrom for the duration of Limsa Lominsa's fight against the Empire as part of the Eorzean Alliance) in exchange for certain freedoms that essentially guarantee the survival of Lominsa's pirate way of life in the future. This is why the Maelstrom has also swallowed, you may have noticed, the official Marauder's Guild.
Thus, we come back to where we started. The Yellowjackets are part of the Navy, the Navy is part of the Maelstrom; the Yellowjackets are now the Maelstrom. Even the old Barracuda banners are now simply Maelstrom bannders. The 'Cudas don't go by 'Cudas like they used to (though some of them do still throw the name around in ARR when referring to strictly-Navy concerns), and, with the Crimson Fleet, Privateers, and Pirates doing a good deal of the fighting, most of the standard Navy is now utilized for keeping order, which you'll notice simply means a huge increase in Yellowjackets. Even Reyner, Lieutenant to Sthalmann and current Commodore, refers to himself simply as Commodore of the Yellowjackets, when you first meet him in ARR.