The difference is more epistemological than practical. Before, 6.022x10^23 was the measured value of the conversion factor between a gram (exactly 1/100 of the Prototype Kilogram) and a dalton (exactly 1/12 the mass of a carbon-12 atom). After the redefinition, a dalton is defined as being 1/6.022x10^23th of a gram, and measured to be approximately equal to 1/12 the mass of a carbon-12 atom. In essence, the dependency chain was reversed, so now it all flows from the kilogram outwards.