Using separate modem and router- public IP changes with new router?

Question 1:

I'm using my own Netgear modem and a separate Linksys router.  Same public IP for years.  Why did my public wan IP change when I switched to a new router?  I kept the same Netgear modem and only switched to a new wifi router.  Doing so will change your wan IP?  Also I put my old router back on and my old IP address came back.  Verified with whatismyip.com  I was hoping changing my router, and not my modem, would not change my wan IP.  

Question 2: 

Can you ping a non-static xfinity wan IP in cmd.exe?  When I try to ping my friend's public IP (xfinity accout) and they try and ping my IP, we get no reply.  We have non-static IP's from comcast.  You can't ping a non-static public IP?

thanks