FWIW, the main tangible benefit I've been able to use is free wire transfers. My daily ACH limit is only 100k which often leaves me needing to initiate a wire transfer. I also use the higher daily ATM withdrawal limit from time to time.
You will get a "banker" who will call you every now and then to ask if you need anything.
Once I got a call notifying me of a CD APY promo that I was interested in. So I stopped by my nearest Chase branch only to wait 15 minutes to be told that since I am Private Client they could not open the account and I had to schedule a meeting at a Private Client branch ...Kind of a hassle for a 1.75% APY.
Note than many private client branches no longer have teller services (although the one's I've been to do have the enhanced ATMs that let you even choose bill denomination and allow your max daily limit of withdrawal as opposed to the ATM's limits)
My bottom line on CPC is it is 95% marketing and 5% free wires and higher cash limits.