Playing songs on my computer via iTunes always updates play counts. No problems there. Just songs played on my iPhone. All my music is downloaded to my iPhone. Play counts don't update whether playing songs straight out of my library or from a playlist from my iPhone.
Yesterday, the local Apple Store told me to enable "Use Listening History" in my iPhone Music settings. Then I played some songs on my iPhone away from home. When I synced my iPhone to the PC with iTunes, the play counts updated. I thought I had solved the problem!! However, that was the only time. Additional plays on my iPhone did not update the play count on iTunes no matter how many times restarted both my iPhone and the PC with iTunes.