Cleaning and polishing by hand, never had any issues with Aramith resin balls, and generic polyester balls, by using a Mister Clean-type eraser sponge and Turtle Wax (paste/semi-soft). Use a reflection of light. Look where the grime and tap marks are. Gently rub like you are shining shoes. Tough marks may need circular rubbing. Do only enough to remove marks and grime. Then wipe with a cloth to get dust off the ball from the eraser sponge. Apply Turtle Wax. Here's where the work gets different. When removing the wax, use one paper towel to remove heavy wax, then a second paper towel to remove the residual secondary lighter wax. Next a good buffing with a cloth, followed by a synthetic micro chamois cloth as the final buffing to ensure no residue will be transferred onto the table fabric.
I play five hours daily, seven days a week. About ten hours to twenty hours of play, will be the timeframe for cleaning the balls over again, as I visually inspect and, pay attention to the performance on the table. As things slow down, it is an indication that ball cleaning is needed.
For old billiard ball sets that are really filthy, it will usually be more of an issue with polyester balls than resin balls. In either case, I use automobile headlight lens cleaner, such as Blue-Magic. Shake the bottle really good, then as I am right handed, put a small amount of that cleaner into my left palm. The hand is a natural shape for accepting a sphere. Roll the ball briskly against the palm, and the brown residue usually from photosynthesis, will come off. It could take a bit of cleaning depending on the condition of the ball. Then with a paper towel, remove the cleaner. Follow that with Mother's Mag & Wheel polish, applying with fingers, and that will slick the ball. Then simply look for any tap marks, and read the top of this response for the eraser sponge and wax application.
Note: Your hands and fingers will be brown/yellow for a couple of days. It will look like the old timers who smoked four packs of cigarettes. Usually, rubbing alcohol and hand soap, will help remove some of the coloration. Cleaning the balls with full servicing, can take 2.5 hours of time, while basic eraser and wax service will take about 1.25 hours. My table has worsted teflon fabric and is a very fast table. When the balls are racked and slid to the footspot, the feel, tells the whole story.