Yes, the Tu-144's nose also drooped to increase visibility while landing. Both the Tu-144 and the Concorde have no horizontal stabilizers and are essentially delta wing (More accurately, it's more of a double-delta. I think their wing designs are called Olgive [sp]) designs. One characteristic of this design is a relatively high angle of attack required for landing*. Given the long noses ahead of the cockpits (necessary for streamlining) and the high angle of attack, the droop snoot was seen as the best way to increase pilot visibility during landing.

Interestingly to us aviation geeks, Boeing's 1990's High Speed Commercial Transport (follow-on SST) was going to use a TV display instead of a droop snoot. This was going to save several thousand pounds and a lot of complexity. NASA even tested a modified 737 with a second cockpit to test the concept. It worked quite well.

*Concorde's wing design was a major factor in the crash back in 2000. The plane also requires a relatively high takeoff speed and relatively high angle of attack right after takeoff. When the plane caught fire, the Flight Engineer shut down an engine below the minimum necessary airspeed to climb with an engine out. That, combined with a downwind takeoff at a weight over limits effectively doomed the plane. According to a recent program I saw, the plane had other problems including a missing spacer on the left main landing gear (maintenance error). Photos taken of the runway after the accident showed the plane was beginning to swerve off the runway well before it hit the piece of metal from the Continental plane. Evidence shows the pilot tried to take off below the minimum takeoff speed because he was about to go into the dirt.