My Arduino projects don't require the power of a 32-bit processor, but do run on batteries. How much more (... or less maybe?) power is drawn by this processor?
It's reasonably close. If you check the datasheets for "Static Characteristics" / "DC Characteristics" you'll find:
The LPC1114FN28 (the ARM chip) draws 9ma @50 MHz, 6ua @deep-sleep, and 220na @power-down;
The ATMEGA168PA (typical Arduino-ish AVR) draws 4.2ma @8MHz, 0.8ua @power-save, and 0.1ua @power-down.
These numbers are just for the chips - the Arduino draws considerably more (about 40ma @idle), and you can stretch your batteries a lot by hacking it. To give a sense of scale, the power LED on an Arduino probably draws 5-10 ma just by itself.
Note that this is a "Cortex M0" profile ARM chip - M means Microcontroller, and 0 means low-end. This is a 50 MHz chip with 32K of flash and 4K of RAM. It's more powerful than an AVR, but don't expect to boot Linux on your breadboard with this thing... that's a job for the Cortex A (Application) series.