Hawk37's explanation is basically it; Vs0 is the minimum speed demonstrated in a stall, and the aircraft can actually be below 1'g' during the manoeuvre with the result that the aircraft wouldn't actually be capable of sustained level flight at Vs0. Consequently it's sometimes called "Vs-min"
Vsr0 is the minimum speed corrected to 1'g' flight - which means each speed is corrected for the instantaneous 'g' and THEN you take the minimum - or, alternatively, you can work out the lift coefficients during the manoeuvre, then take the maximum, then calculate Vsr0, because it's basically the same thing. The correction to 1'g' is why this is sometimes called "Vs1g".
The reason the regs have changed to the Vs1g method is that its inherently more consistent - depending on how you fly a stall you can get different Vsmin values, while Vs1g is more analytically defined. It's much harder to 'cheat' (intentionally or inadvertently) at Vs1g! However, a Vs1g speed is invariable higher than a Vsmin speed FOR THE SAME MANOEUVRE because the 'g' was invariably below 1.0. Therefore had the same 1.20/1.30 factors been retained for V2 and Vref, these speeds would have increased, with a consequent performance (and economic) penalty; since those factors are arbitrary, the argument was made (and its logical) that the old factors, combined with the Vs-min method, gave a "safe" V2 and Vref, and therefore the goal should be to end up at similar speeds with the new method - so-called "equivalent safety".
It was determined that for then-existing aircraft that a Vs1g speed would be about 6% higher than a Vs-min speed, so the factors were all reduced accordingly - so-called "reduced reference speed (factors)" - 1.20 became 1.13, 1.30 became 1.23, etc. For any given aircraft one method or the other might give 'better' V2 or Vref numbers; the main point is that the Vsr0 number is consistent.
(Incidentally, there were minimum 'g' levels for a valid Vs-min stall speed - you couldn't fly a 0.5'g' 'bunt' and claim that as stall speed, but values down at 0.97-0.95'g' were certainly seen in some cases)