The uMP3 has been around since 2003 and is tried, tested and true. It's original design was intended for OEM integration. It uses the ATmega32 controller running at 7.3728 MHz.
The rMP3 can also be used for OEM integration, but is also an Arduino compatible shield. The controller (ATmega644P) is running at 2.5x the speed of the uMP3, enabling more processing in the same amount of time. The larger flash size on the controller also provides more space for firmware improvements.
They both have the same base function set (playback controls, external triggering, config on card (COC), file access, etc). You will see more improvements on the rMP3 versus the uMP3 because of the flash constraints.
You can use either with an Arduino (libraries work with both), but the rMP3 can piggy-back directly onto an Arduino:
