This is just my two cents as well, but I'm able to hear it. In fact, I did a check right before writing this. Woke up an hour ago, didn't do any audio work yesterday so ears are fresh.
In short, most of the difference I can hear is in the upper mids to highs, as far as just the API 2500 bypassed off and on.
Also FYI, the saturators he uses on the Kick/Bass Buss have nothing to do with tape. The Waves (Nevo) setting is emulating the The Neve 5116 console.
This is not an entirely accurate statement.
Watch and listen between 7m40s and 8m20s on video 06. Within that 40 second time frame he is bypassing the correct plugin while setting the 2500's threshold.
I am going to guess you have no idea how expensive all this used to be before the software existed.
Waves is probably the cheapest developer out of all of them in terms of cost. The API 2500? The hardware unit of that currently costs $3,388 (including sales tax). The Waves version currently cost $70 and that's when it is not on sale. When I bought it, I got the API collection including the EQs for $70. -- That is a steal compared to what most other developers would charge, likely $300-400.
-- In a way, I slightly envy James for being 25 years old, and having all of this available. Back when I was still 25? FL Studio was called FruityLoops, and none of the software instruments used in this tutorial existed...and neither did computers fast enough to run them without crashing... you wanted decent sounding plucks, leads and pads? You needed the hardware, the rack space for the modules, studio space for the 49+ keyboards and hardware mixer, the cables to interconnect and of course the budget. -- Wanted the sound of an SSL 4000? One needed an actual SSL 4000. -- Count your blessings.