Maybe this helps:
https://ad.easa.europa.eu/blob/SIB_2.../SIB_2009-16_1
As far as I understand one expects a specific rate while the other prescribes rates that must be avoided.
From the linked document:
"Monitor Vertical Speed” RA always requires that the
vertical speeds prohibited on the RA display are adhered
too. Typically, a “Monitor Vertical Speed” RA does not
require a change in vertical speed.
“Maintain Vertical Speed, Maintain” RA always requires
the climb or descent to be maintained in accordance with
the indications on the RA display
“Adjust Vertical Speed, Adjust” has been replaced by "level off" in TCAS Standard 7.1 (if I remember correctly).
As far as daily workings as a pilot are concerned, I am with Stilton: KISS (keep it simple stupid).
Stay in the green and out of the red - which is a lighter way of saying what the official TCAS standards say: "never manoeuver against a TCASRA".