Hangtime would still play the same no matter what joystick you used. NBA Showtime I think was 49-way, so that game would benefit from an analog stick.
A 49-way is different from an optical analog joystick. A 49-way is technically a digital joystick, but reads 49 different joystick positions. Unless you are looking restore a game, 49-way is falling out of use. If you get a GPWiz49 from GGG, you can breathe some life into 49-way joysticks turn them into a versatile joystick if you have some 49-ways on hand. You can easily handle 4/8 way games on it as well.
Optical just means that it uses optical sensors to tell the position of the joystick. There are different types of analog sticks out there. Other analog sticks can use magnets or potentiometers to determine the joystick position. Any analog joystick is gonna have a better "resolution" than a 49-way joystick (although most games available in MAME won't take advantage of that). Right now, most people looking for an "out of the box" analog stick for their setup will get a U360 from Ultimarc. It has a built in encoder and you can custom map it to work as a 4 way or 8 way based on the game you are playing.