What the thickness actually does
A thicker core absorbs more of the ball's energy before returning it. That makes the paddle quieter, softer and more predictable, and it widens the area of the face that behaves the way you expect. The cost is that you have to supply more of the pace yourself.
A thinner core returns more energy. The ball comes off faster for the same swing, which is free power on a drive, and it also means off center contact is punished harder and resets take more skill to keep down.
How to tell which you are on, and which you want
Most players moving from thin to thick describe the paddle as becoming quieter and more predictable. Most going the other way describe more free power and more unforced errors. Neither reaction is wrong; they are the same trade seen from two directions.
If your misses are long, with drives sailing and resets popping up, a thicker core will usually help more than any change in technique you can make this month. If your problem is that balls die into the net and you feel like you are muscling every shot, a thinner core is worth trying.
If you have never thought about it and do not have a specific complaint, 16mm is the safer default and the reason most of the market sits there.
Buying either one secondhand
Core thickness has no bearing on how a used paddle wears. The face and the perimeter are what age, not the core dimension. So this is purely a fit question, which makes the secondhand market the cheap way to answer it.
That is the real argument for trying a thickness used rather than new: you are testing a hypothesis about your own game, and being wrong costs a fraction of retail. Every listing here states the core thickness, so you can shop the specification directly rather than guessing from the model name.