New-World started with the PowerBook Lombard G3 and all of the post-beige, candy-colored Jobs machines from the bondi blue iMac, blue&white G3 tower, graphite G4s, titanium and aluminum PowerBooks, etc. When the Mac boots, the ROM is read into RAM. Instead the ROM is a file in the System Folder which contains information about the machine.
New-World Macs were based around the technology of 'ROM in RAM', which did away with ROM chips. up until the final ones, the beige G3s and PowerBook Wallstreet. Old-World Macs are the ones that are usually beige or platinum in color (off-white) and have letters and numbers for names, ie: SE, IIfx, Quadra 950, 7200, 8500. Old-World had actual ROM chips inside, but as Macs and the OS became more sophisticated, it was getting harder to fit everything on there.
This is what the difference between 'Old-World' and 'New-World' Macs is about.