Well, it'll use more power than slower response time for sure. As for how much more, it depends really on the resolution and many other factors. The higher the resolution, the harder the controller has to work in order to change each individual pixel fast enough. As it is, the current screens are already battery hogs when HDR is played back or when 120Hz mode is engaged. Would you be okay with 5% less battery run time?
That's not how it works. 120Hz means the screen is updated 120 times per second and it is going to be updated 120 times per second. What this "response time" thing means is that when an update is initiated, the previous frame has not finished changing yet. This creates an "in-between" effect that results in motion blur, as frames are blended together.
Motion blur is bad for use cases that require clear in-motion images (like gaming, or artists working on high framerate animations), but it shouldn't be a problem for many other industries/disciplines. For instance, most video cameras are still around 24-30fps, and this response time should be fast enough for that content, so there's little to no "motion blur" when playing back 24fps or 30fps videos.
What's the added value? 120Hz is just... smoother than 60Hz, which is smoother than 30Hz. Have you seen the iPad Pro display? This is like that. The iPad Pro is also not a fast display, but it still gets smoother when running at higher refresh rate.