the sound made with your teeth (usually when you are angry) in English is "gnashing your teeth" or "grinding your teeth" [rechinar los dientes] and you are correct, you usually hear that rather than seeing it.
"Gritting your teeth" is an action where you clench your jaws and hold your teeth together [apretar los dientes], usually when you are scared or worried. In this case there is nothing to hear but people may see it (because it's usually like a grimace [la mueca ??] where your lips are apart).
I hope that helps. (Sorry that I can't translate more of it into Spanish.)
Conejillo (nice nickname, by the way!):
From what you've just told me, I deduce that he "mantenía tensa la mandíbula", or something like that. I'm not pretty sure how will I translate that, but I think it will be something related to "tenso", or "tensión", and maybe not using the word "dientes" but "mandíbula".
that sounds good, but if I hear that someone is "gritting their teeth" it also conveys the idea that they are either in actual pain or that they are scared/worried that something bad is about to happen.
For example, if I'm stuck in traffic I may "clench my teeth" (or my jaws) because I'm tense or frustrated, but if I have to slam on my brakes to avoid hitting the car in front of me I'm likely to "grit my teeth".
Another example, if I'm in the Doctor's office and they keep me waiting for a long time, I may "clench my jaws" or "grind my teeth" because I am frustrated (or impatient), but when the nurse sticks the needle in my arm to give me an injection I will "grit my teeth" so that I don't cry out. (I am gritting my teeth because I know it will hurt but I want to look brave.)