iTunes isn't downloading updated apps from iPhone

Is your phone on iOS 9+ ? If it is then apps won't copy back from it, there was a developer function added in iOS 9 called 'app thinning' that allows only what is needed by an app on a device to be downloaded onto that device, so you may not have the complete app (what would be downloaded on a computer's iTunes) on your phone . To get the updates on your Mac you will need to download them on your Mac.

With iOS 9 and app thinning you might not have the complete app on your phone, only what is needed for it to work on an iPhone (and potentially, I assume, that model) - which might be ok if that iPhone is your only iOS device.

I download updates on my computer's iTunes and sync them from there to my devices