I think the first important thing is having a matching set. I've seen it a number of times in bars. The cueball is a different weight than the other balls and the cueball can either be overeager to draw or sluggish to draw. Or I've seen object balls that don't match where some are shiny and others are dull...some have numbers on the stripe and some have numbers off the stripe. That shouldn't matter until you go to rack the balls and find out they are of slightly different sizes and you cannot get a tight rack.
After that for me is durability. The older the balls are the more likely their surface starts to get worn and abrasive. That also gives more places on the surface of the ball to collect crud like chalk, increasing the number of times a shot might "skid" meaning the balls can have a high friction rub against each other during collision causing the object ball to throw more than usual on a cut or causing the cueball to climb the object ball slightly and dull out the spin you put on it. An informed player will be annoyed that the shot outcome was aversely affected by these types of conditions. An uninformed player could be confused thinking they did something wrong when it potentially wasn't their fault.
After that for me is the aesthetics. For me, I mostly appreciate familiarity. I prefer to practice in conditions similar to where I compete...and vice versa. Being in a traveling bar league, I'm obviously not seeking to bring bad playing conditions to my home table. But I would prefer to not own TV colors, marble elephant balls, or any other schemes that wildly differ what I'll likely encounter in competition. Some things like that can matter more to some than others. I bought a new set of balls to bring to league. It was interesting to see how many players commented on being unnerved playing with balls that were so shiny. Many different players from different teams over quite a few weeks made comments.
After that it's like picking a cue stick. Someone will say one stick has a good hit, or feel, or sound, or reaction, etc. over another stick. Then someone else will come and say something very similar but thing the other stick feels better. Then someone will go around telling everyone all the intangible reasons they think one is better, then other impressionable people will start repeating those anecdotes to others. Obviously there's a ton of faux experts in pool trying to raise their street cred by emphasizing something insignificant or pat out false. Similarly with balls, I just don't put much stock in those things. Generally speaking I think just about all the balls (in matching sets and good condition) will sound and react good enough for me to comfortably hop on a table, adapt, and have fun playing pool.