Sure they do. After all, light is light.
But the thing to consider is the output of light and how that will affect your ability to make the exposure that you want.
When taking photos in ambient light, you rely on the shutter speed, the aperture and the ISO to get your exposure. If you use constant type lighting, it's exactly the same. Now, if you want to get sharp photos, you must control the length of the exposure, especially if either the subject or the camera is moving. So when shooting people and/or holding the camera in your hands, you must maintain a decently fast shutter speed to get a sharp photo.
So going back to our exposure triangle, if you need a fast shutter speed, you may have to compensate for that with either a larger aperture or a higher ISO. A larger aperture will give you a shallower DOF, which you may or may not want. But you'll also find that you reach your lens's maximum aperture pretty quickly. You can raise the ISO, but the more you do that, the more digital noise you will get. It's a constant compromise that may not be to the benefit of you photos.
Of course, the brighter your light source, the easier it is to get that faster shutter speed...and that's the problem. Most constant light sources, really aren't that bright. You could use something big & bright, like stage lighting, but those create so much heat that they aren't great to use for portraits etc. Regular household lamps and probably LED arrays, probably don't put out enough light to get the shutter speeds that you want, while allowing you to use a lower ISO. I'm sure that some people are doing it, but it may take a lot of LED lights to pull it off.
Now, if your subject isn't moving, and you put your camera onto a stable support, then you can use just about any shutter speed and still get sharp shots, so any type of lighting should work.
So now we can talk about flash. When using flash, the burst of light is really fast. Probably a lot faster than you shutter speed. So because the burst of light is so fast, it freeze the motion of the camera and/or subject...allowing you to get a sharp photo. And actually, the shutter speed doesn't matter at all, in terms of the exposure from the flash. That's why flash/strobe is the choice of so many photographers.
Lastly, keep in mind the color temp of the lights you are using. You can change/set your White Balance to match your lighting, but if you have more than one type of light source, you may not be able to match that with your camera, and you'll end up with a color cast in your photo.