Want to charge fast? Use the included charger!
The charger looks identical to the one provided with the S4 and Note II, it's 2A after all.
However it isn't! Galaxy Charge Current Lite (in the play store) shows 1200mA when plugged into this charger.
However the charger included with the Note 3 will show 1800mA.
The difference is the output voltage. The Note 3 charger is 5.3V whereas all the others are 5.0. This slightly higher voltage can allow the phone to charge at the higher current even if your cable has a bit of voltage drop. I have not tried with other cables however.
The (Note 3) does seem a bit more sensitive to charger type and will drop down to a lower charge current faster, however.
My 10Ah Anker bank will charge at 1200mA, for example. The 2A car charger I have also charges at 1200mA.