The main advantage of continuous lighting over studio flash kits is that with a continuous lighting setup what you see is what you get. The composition you set out before you is what you will see as the final result in your photograph, whereas with flash lighting, you don’t get to see the final result until you’ve taken the photograph. This can be particularly advantageous for beginners.
Another advantage is the ability to shoot a wider apertures for a shallower depth of field look with can be difficult with flash lighting because of the higher light output.
The third advantage to using continuous lighting is that it can also allow you to shoot video ideal for a combined video and stills shoot.
So should I replace my studio flash kit for a continuous lighting setup? Well, no would be the short answer.
Flash can still be a useful tool in your lighting arsenal, especially if you want to freeze motion in your photographs which is where flash has one big advantage over continuous lighting. It’s far more difficult to freeze motion with a continuous lighting setup than flash and to do so would involve either opening up your aperture or increasing your ISO to compensate for the faster shutter speed, whereas with flash lighting its far easier to achieve using the duration of the flash to freeze the motion.