Not to get into a science lesson, but my $0.02:
Breathing compressed air at depth has 2 main negative effects. The first is of course is Nitrogen loading which dictates your time at depth, and the second is Nitrogen Narcosis. Many divers experience Narcosis at about 100+ feet, and diving with a training course will expose you to Narcosis under controlled conditions and allow you to see how you will react.
From Open Water training remember Boyle's Law. As you dive deeper, your air consumption becomes a multiple of your depth (in Atmospheres) times your Surface Air Consumption. This makes monitoring your air supply and consumption even more critical!
Finally - deep diving requires a greater reliance on your buddy and your gear, and starts bringing factors into play such as gear redundancy, self-rescue, and gas management.
A Deep Diver Specialty (with any of the fine agencies) allows you to gain the knowledge and experiences necessary in a controlled environment to handle these situations and understand the risks. It's not about the card.
As to the poster who related stories about Open Water checkout dives on the Mighty O, many agencies - such as SDI - limit Open Water Training dives to a maximum depth of 60' (for many of the reasons noted above). I would not want to use this for a student's first Open Water dives with a hard bottom waaaay down at 200'+, and a new diver with less than perfect buoyancy!
I'd go for the Deep Diver training!
Dive Safe!