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I bought a Player and just upgraded it (Hipshot Tuners, Bridge, elec),and it was still less expensive than American models. I like the skunk stripe, the Truss adjustment at the Headstock ( even has a plastic protector there). After i upgraded seems like the difference is minimal. CNC is used on both. Maybe you have to try them first though.My opinion, buy the Player II for the value it offers, buy the American Pro II if you want something better.
I currently own two US made basses and two imports. They all sound and play great! The US built basses (and guitars) I’ve owned all share one thing in common and it is tuning stability. My Fender American Pro II P5 and my Rickenbacker 4003 both hold tuning for weeks, inside or outdoor gigs require no retuning during sets. Just my experience and if I were in the market for a Fender Jazz it would be the American Pro II.