My Motorola Droid Razr draws up to 1.3A from the USB charging port (assuming the port can provide that much current). So a regular laptop USB port or an off-brand charger may not be able to charge the phone as fast as the OEM Motorola charger.

The current drawn from the port does not change when the screen is locked or unlocked. It seems to depend mostly on battery charge state.

I have observed that when the battery is completely discharged (and the phone won't turn on), a laptop USB port (or an off-brand charger I normally use when the battery is not completely discharged) is unable to charge the battery. The symptom is that the front LED turns on white and stays like that. If the phone is turned off while on the charger (long press on the Power button and the volume down button at the same time for 10-15 seconds), then the phone seems to be charging.