the difference between a drum kit and a drum machine is that a drum kit you perform naturally with your hands using sticks to change the pitch on whether it is high or low, and a drum machine is like a little electronic board where all you have to do is choose which percussion instrument you want and edit the way the rhythm sounds. personally i prefer the drum machine, much easier ;)
a drum kit is an instrument that you can play. a drum machine is a computer that generates a drum track. drum machines are commonly used in techno and hip-hop.