What Do You Hear When You Adjust VTA?

I tuned it to the sound of a Cembalo in the midst of an orchestral Bach suite with excellent success. On an Eterna digital recording no less. The VTA was right, when the cembalo sound became clear and airy and the soundstage opened up both in depth and width.
What Viridian has mentioned is exactly the side-effect of a properly set VTA in my experience.

I hear a shift between muddy slow plodding muffled sound (VTA too low)- to shrill, hurt the ears, ringing nastiness.
With my line contact stylus the correct setting is quite precise and the range of acceptable is narrow.

This is an interesting but very controversial subject as it relies a lot on what people hear and how they describe what they hear. I am not saying VTA makes no difference but your great question will be answered correctly by a lot of people who sound like they are disagreeing. I know it makes a difference to get it right and once you hear that sweet spot, you will understand. In my experience it is like the music becomes distinct notes with beginnings and endings. The sound stage or spatial impression of the different instruments becomes more apparent. There is a smooth transition from treble to bass without ringing(tinny sound) or slurring particularly in the bass. What is hard about this is that some cartridges seem to have a very small "sweet spot" versus others. The other thing that one must tweak when you move VTA is the VTF as they play off each other to some extent. (IMO only). My experience with the ZYX Universe is that a VTF on the lighter side of the acceptable range with a slight tail down is close to the sweet spot, then slowly move up a small amount, then move down over a listening session. One small tweak is to wait until the second or third record to do serious adjustments. I do believe it takes a couple of records to get the suspension loose. Also on cold days, it may take a little longer if you let it get too cool.

I agree, again.
One thing that has helped me a lot is to play many records within a boxed set. Then you can be pretty sure the pressings are more or less uniform in vinyl thickness and the cutting angle. One set I have has aprox 100 records in it of all different periods in classical music. You get harpsichord - very difficult if the VTA is too high - strings, organ music -plodding, poor muffled bass if VTA to low, voice (opera) which sounds a mess if VTA either way is incorrect.
Pretty quickly you can dial it in.