So apparently it's all over the news now that "Seal Team Six" was responsible for "Operation Geronimo", the raid to take down Osama Bin Laden's compound. I first learned about ST 6 while reading Richard Marcinko's biography, "Rogue Warrior". Marcinko was the Commander who established ST6 as the Navy's specialized counter-terror force, and created "Red Cell", the Navy's specialized faux-enemy infiltration unit, created to test US Naval facilities' ability to defend against elite hostile penetration teams.
So apparently things have changed a lot since the last time I did some digging. Here's the Wikipedia article on the US Navy SEALs, their history and organization. Notice that "Seal Team Six" isn't part of the organizational chart anymore; instead, it has been formally replaced by the US Navy Special Warfare Development Group.
"DEVGRU" is one of two "Tier One" Joint Special Operations Command operational units, the other being the more famous 1st Special Operations Detachment - Delta, aka, "Delta Force". Here's the article detailing the Tier One / Tier Two breakdown; I don't know if this is a truly formal designation or just a naming convention that gets used to separate the Black Ops forces from the more public special operations units.
So to me, it appears now that ST6 / DEVGRU stands apart from the conventional SEAL teams in the same way that Delta stands apart from the other Green Berets - members are recruited from those parent units, and are still ostensibly part of those units, (i.e., a ST6 member is still a "Navy Seal" and a Delta operator is still a "Green Beret") but operationally, and I imagine to some extent culturally, they stand apart. I knew this was the case for Delta, but I wasn't aware that ST6 had moved so far away from its association with the other Teams.
EDIT: See Jack Murphy's comment below regarding Delta membership. Thanks dude!
So, there we go. This is what we learn when we're bored at work and haven't been doing our SOCOM homework lately...