I flew Concorde just last month from London Heathrow to New York JFK. It was a charter deal which included economy return in a BA 747-200, four nights at a Sheraton in Manhattan, a bus tour, a boat tour and a helicopter trip - all for about 2800 GBP. Although it was a charter, the Concorde service was the full deal.
To be honest, there isn't much evidence you're in such a special aircraft. The seats are more comfortable than economy class and big displays at the front of each cabin show you how high and fast you're going and the acceleration and flying at Mach 2.0 don't feel different to a normal flight.
But if you're an enthusiast and read up a bit on the unique operating procedures before you go, then you just know how special your flight is, especially when the reheats kick in again, two at a time, as you begin to accelerate through Mach 1.0 and again when those displays show 1290 mph at 58,000 feet. A trip to the cockpit during the cruise lets you see the unique view through the double windshield as well. And fillet steak for breakfast was a nice change.
One tip - normally I prefer not to sit over a wing but on Concorde try to get a seat about halfway back so your view isn't completely obliterated but you get a reminder that you're not flying any old aircraft every time you look out of the (tiny) window. I was in row 7 so looking out of the window was just like looking out of any aircraft.