Fair question. I realize this thread has gotten messy, so its understandable if no one goes through and re-reads everything. I'll try and keep the restatement short and concise. The big guy's don't compete on sound quality. They compete on project capacity and speed. Bigger consoles allow multiple machines, multiple operators, and different types of data to sync to a central nucleus of control.
Below is the Harrison console at Sony Pictures.
Take a look at how the 3 mix stations are set up. Multiple DAWs spread across six or seven computers, all feeding the console. (The A and C mix studios at Disney use the Soundmaster Ion Machine Control system to keep everything together). No one person can manage this kind of job with a single DAW and a mouse, because of the amount data and speed operators need to mix it at is insane.
This is my console...its the System 5.
It can scale to over 1000 channels that any combination of digital and analog. It has onboard DSP which means it has a computer independent of the DAW, yet it can control every feature within a DAW. All mixers have a 2 bus (stereo), and some like an SSL 9000 series have a surround bus (dolby 7.1). The Harrisons and the Avids have 64 channels on the master outputs. They have fully automated recall even for the analog side (including gain staging, aux sends and routing). Most common mixers have 6 or 8 sends and returns. These have 256 sends and returns that can routed with a mouse click, eliminating the need to ever touch a patchbay. The System 5b models like NASA, Oprah, and Jay Leno used (I own Jay's old one) have a redundant emergency changeover systems. Another thing they do that DAW/itb can't is tie directly into a multi-user media server to access and edit files stored on an enterprise network without having to move a single folder to a local drive. Avids current enterprise storage system is called the Nexus...the older generation was the Isis.
If you were curious, those are a few things that are a little different than a normal mixer. It's a good question to ask!