In essence, Shutter Speed is the same thing regardless of whether the camera shoots photos or video. However, there’s a difference in how we use Shutter Speed.
Photography
In photography, most often advice goes on getting a sharp image and thereby shutter speed is a component in getting the right exposure. My primary advice was always to shoot double your focal length. Say you shoot a 15mm then your shutter speed should be at least 1/30. That said, I never went below 1/100 as that as a rule of thumb secures you a sharp photo most of the time.
Videography
Now, in videography, we need to think of shutter speed slightly differently. As we are introducing movement into a shot, it’s no longer about freezing the moment as it is in photography but rather capturing it.
So, to get the shutter speed right in videography the rule is to shoot double your frame rate, meaning; if you shoot 30 FPS your shutter speed needs to be 1/60. If you slow motion, say 60 FPS or 120 FPS your shutter speed needs to be 1/120 or 1/240 respectively.
Now, if you were to shoot at a higher shutter speed it’s not that it breaks your footage but you’ll start to freeze each frame more, getting everything sharp and thereby not getting the motion blur that our human eyes are used to seeing. Without the motion blur, the shot will look off and while it can be difficult to pinpoint what’s wrong, that will be it.
So to recap
In photography, depending on style and purpose, we can shoot with a super fast shutter speed to freeze moments in example sports, we can shoot with a double the focal length shutter speed to ensure sharp photos and help exposure for say portraits and a slow shutter speed for long exposures.
In videography, we always (as much as possible) want our shutter speed to be double our frame rate to get the right motion blur in our image.