Arcade sticks are good for fighting games like Street Fighter and Tekken. Fighting games usually require intricate directional movements and rapid button presses. You can accomplish such actions more easily with an arcade stick that has a joystick with big buttons laid out conveniently.
Since controllers have shoulder buttons, it can be awkward to do a combo with a mixture of face buttons and shoulder buttons. Arcade sticks usually have all face buttons, and the other buttons are often secondary buttons for pausing the game etc.
Admittedly, games like Mortal Kombat may be better suited for controllers and not arcade sticks, but generally speaking, arcade sticks work fantastically for fighting games.
Of course, arcade sticks can work with other genres as well, but make sure they do not require two analog sticks. Otherwise, feel free to check out our top arcade stick picks.