my chinese made trex clone produces an insane amout of lift. after the lipo pack i was using decided to up and die, i didnt have money to replace it, so i gathered up every nicd, ni-mh, and even a few non-rechargable aa cells, taped and soldered them all together into an equivalent battery pack weighing about 2 pounds. and taped under the skids. it took off and flew for about 2 minutes before i crashed it. brushless motors pack a lot of power in a tiny and lightweight package. that first video i linked to for the manned electric octocopter you wouldnt think it capable of flight at first glance. companies have been trying to use the tech for all kinds of maned flight applications, namely planes for pilots with sport class licenses, but the faa really doesnt want to see cutting edge technology in sport class planes, and would rather stick to the tried and true lawnmower engines.
older gas helis had weight issues and the engines were not all that powerful for their weight. plus you needed to carry fuel in addition to batteries (usually heavy nicd cells) for the electronics. servos were bulky and no way around it you need at least 4 in a heli plus another to throttle the engine for gas helis. and you could have a gyro, which were heavy mechanical units. these days you have little 9 gram servos, escs are tiny and light weight, a single power system for everything, solid state gyros and a receiver smaller than a matchbook, leaves considerable margin of error as far as weight is concerned. quad copters likely have even more margin for error. 4 lifting motors, no servos (just escs), and you do all your stabilization and calibration with electronics (a regular heli needs to be mechanically calibrated as well as electronically) and in software.