Nothing wrong on the phone or cable. S4 and note 3 have larger capacity batt, it can't really charge by laptop usb port, only ac power can charge the batt. It common to see the batt level go down when it plug to usb port at the laptop.
Maybe apple is aware of the issue and have not intention to go for larger capacity batt for iPhone, my 2 cent of thought..
Eh not true for the Note 3. My desktop's USB 2.0 port charges it fine, though it takes a damn long time at 500mA vs a USB 3.0 port's 900mA vs the AC charger's 1200mA. Put it this way - you can fill a reservoir with a tap, it just takes damn long. It takes roughly 6 hours to charge it via USB 2.0 than the 2.5 to 3 hours via the AC charger.
But if you start playing things like PvZ2 on the Note 3 while charging it via a USB2.0 laptop port, then the power draw might exceed the current supplied by the port, so the battery level will still drop.
If you plug the Note 3 to an unpowered USB port though, don't expect it to charge of course. Those things provide something like a 5mA current.
I had the problem of falling battery levels while charging before on an old HTC Desire S. It's either the battery's fault (not likely for the Note 3, but you never know), or the USB cable's fault. You can try this app:
https://play.google.com/store/apps/details?id=com.abmantis.galaxychargingcurrent.free
It allows you to check the charging current when you plug in the phone. It should read 1200mA when you plug it into the AC charger.