So not true :o) there's constant research in this field going on all over the net (usually known as "flamewars"). Here's my pick:
General rules
More engines means more power! and more lift. That means more batteries. That means more time in the air. Brushless electrics like to run slow (lower RPMs), so bigger means more efficient. Also, if efficiency is your goal, coaxial = evil.

The breakdown:
Tricopters:     cheap, easy to build, least stable, not as robust (tail servo and mechanics), low lifting power and flight time (because the motors have to run faster to hold it all in the air). No engine out capability.
Quadcopters:     mechanically simpler than tris. While they weigh almost the same they have about 1/3 more lift, they are usually more stable (no servo issues) and are capable of staying airborne for a little while longer (they can either lift larger batteries or fly more economically because the weight is spread across 4 motors and not just 3). Still no engine out capability. If it fails, it goes down.
Hexacopters:     All the good things that quads have, plus more power and more lifting capability. As a bonus they add limited engine out capability - a hexacopter can lose any single engine and still land (it will lose yaw control though), and if it loses one or both engines on the neutral torque bar it could even continue flying unaffected. Downside is that they are larger and a little pricier, especially if you're running high-grade motors like AXI.
Octocopters and heavier:     All the good things from hexacopters, plus true engine out ability. Loses any single one and still flies fine. This is what you fly if you need horsepower and reliability in one package. This is what you strap that $1300 Canon 7D under :)) Even more expensive though. Also heavy craft are really power hungry and unless you have some serious chargers at hand they require a lot of work on the ground before a flight can be made (charging say 5 packs for 25 minutes in the air).

Flight time - a hexa will be heavier and therefore draw more amps to stay in the air. It will however be more efficient, i.e. it will use less watts to lift one kg, so it could afford to lug a heavier battery and in effect stay in the air longer. However with the same battery as a quad it would fly a shorter amount of time.