Some differences you will see in 'somewhat challenging' circumstances, some you might not notice unless you bothered with a formal comparison!
Flare outdoors: when sun is casting light directly onto the filter surfaces, you can tell a cheap filter with cheap coatings vs. a quality filter with quality coatings
Flare indoors: when you have a bright light source seen by the lens within the Field of View, against an otherwise dark background, you can tell a cheap filter with cheap coatings vs. a quality filter with quality coatings
Distortions: I have seen a polarizer acting like a PRISM, and as you spin the polarizer the scene shifts around!
Effectiveness and loss of light: Polarizers use a 'foil' to do the polarization. Foils come in different qualities of effectiveness, and some foils can cause greater or lesser amounts of fundamental attentuation of light (even ignoring the polarization effects)
You can look at test results and see that foils vary in their ability to cut of undesired light.
https://www.lenstip.com/index.php?art=139