The easy part to answer is the part regarding performance and GPU utilization. If your card is capable of 200fps and you cap at 120fps then the card will not need to run at maximum GPU utilization and therefore will draw less power, run cooler and run more silent.
The harder part to answer, is if capping FPS to refresh rate is unoticeable or not. It seems intuitive to say no it wouldn't be noticeable, you have 200 frames generated per second but only 120 refreshes can be displayed per second. But that's not quite how it works. The reason is because monitors do not update the entire screen instantly. They draw each new refresh in horizontal lines moving down the screen from top to bottom. And if the frame buffer is swapped during this process to introduce a new frame then you get what is called a "tear line" which where the old frame ends on some row of pixels and the new one begins. And so 1 full refresh of the screen can be potentially many frames stitched together.
Now you can argue that this is still just 1 frame, sure. But the patchwork of frames is a temporal thing, the rows of pixels closer to the top of the screen are older, the ones near the bottom are newer, they're from a more recent frame. My subjective experience of this is smoother and more responsive gameplay, especially when playing fast paced first person shooters where there's a lot of fast turning. However I will add a caveat, that there's diminishing returns on frame rate. I have a 60hz monitor and so running at say 100fps feels smoother to me. But on my 120hz monitor running at say 200fps doesn't really feel much smoother.
The TL;DR advice would be that a 120fps cap on a 120hz monitor isn't going to diminish your subjective feel for the gameplay by an appreciable amount. But the same probably can't be said for lower refresh rate monitors. I would also say that if you're going to cap, don't just cap but enable vsync so the frames are synced up with the refresh rate and that eliminates tearing. Vsync introduces some latency but it's a very small amount at 120hz