I just finished my AOW through PADI - but I have actually done a number of specialities through TDI SDI - including the Deep Speciality.
The biggest difference was that with the AOW adventure dive, we didn't go down as deep. It was limited to theory, a deeper dive than you'd see in OW, and the demonstration of color fade and narcosis (none of the divers displayed symptoms). Followed by an SMB blow and a mandatory safety stop - you missed it, you failed.
SDI's deep water speciality required, if I recall correctly, two dives. The first to about 100 ft and was pretty much the AOW dive. The second deeper dive (on a different day). On the second dive, was more intense. I recall I was required to carry a pony bottle, and the instructor 'simulated' an emergency by shutting off my air at a random point in time at depth (knew it was happening, just not when), requiring me to signal an out of air situation, switch to switch to the pony bottle underwater and execute a controlled assent with an SMB blow at safety stop depth.