Joysticks are a specialized peripheral for gaming, so the advantages and disadvantages are specific to certain game types and styles of gamer. Some advantages include nostalgia factor and ease of use for handicapped gamers. Disadvantages include lack of compatibility and game options.
Joysticks are most often used for flight simulators and fighting games. They allow for a lot of movement that is easy for the brain to comprehend. Because of the style of games joysticks are usually used for, there is a sense of nostalgia that comes from using one, particularly for people who have a history of playing arcade games.
Many joysticks also have a lot of customizable buttons, making them a great option for people who only have the use of one hand or who want to fight things like carpal tunnel syndrome from using a mouse and keyboard setup. The extra buttons are also a convenient, easy-to-use alternative to controllers.
Many modern games outside of the genres listed above do not have joystick support. This makes it difficult to find a compatible joystick for the majority of games. In addition, this means that there is a limited number of games for which joystick setup is simple. It also means that purchasing a joystick means purchasing an accessory that is only easily usable for a few games.