OK, I'm going to elaborate slightly as I think the real difference has been missed.
LIM stands for Linear Induction Motor. The name is already giving an indication as to the way these work. Basically you have a series of electromagnet plates placed along the track, attached to an AC current source. As the magnetic field is induced in these plates they generate an opposing magnetic field in the fins attached to the train (via eddy currents - basically the alternating magnetic field causes a current to be generated in the plate on the train, which then itself creates a magnetic field) and Lenz's Law states that opposing magnetic forces will repel each other - thus pushing the train along. Basically you control the speed of the train down the track by alternating how fast you change the frequency of the AC current.
LSM stands for Linear Synchronous Motor. This means that the fin attached to the train is a permanent magnet, and so is affected by the generated magnet field directly. All you have to do it charge one of the electromagnet plates behind the train to be the same as the back end of the fin (the front end is charged the opposite - like all dipole magnets). This then pushes the train along. In reality, they also charge the electromagnet plates in front of the train to simultaneously pull and push the train along. All you have to do is keep the pulse just ahead/behind the train all the way down the track.
So, what's the practical difference?
LIMs are simpler (that meaning the hardware to increase the frequency of an AC current isn't that complicated).
LSMs are more complicated (you have to have a clever control system to monitor the exact position of the train to ensure that the magnetic field is in the right place).
BUT
LIMs are generating currents inside the fins on the train, which causes heat to build up.
LSMs are capable of braking the train on the way back without any power. As the permanent magnet fins pass the plates in the launch track, they induce currents in the plates and (with eddy currents again) this generates an opposing magnetic force which slows the train down. You can't do this with LIMs because without any current flowing in the plates, it would just be two sets of metal plates passing each other. LSMs work in braking just like normal magnetic brakes on a coaster do. It's worth nothing that this effect is a fuction of several things; magnetic flux, distance between plates, etc, but crucially - speed. This is why you can't ever stop a train directly using static magnets. The train can slow down greatly, but you'll always notice some sort of friction device to actually bring the train to a stop*.
FINALLY on the topic of power consumption, I don't know this for sure but this is my educated guess. LIMs are activating the magnets down the whole launch track at the same time, which takes a lot of power. LSMs are only generating magnetic fields in the vicinity of the trains, and so don't require as much power (I don't imagine the more complicated control system uses up more than the difference), so that might be another factor which means LIMs are becoming less popular these days when compared to LSMs.
I think that just about covers it, I might have missed something, or messed up a detail or two, but that's about right.
*The topic talking about the Inverted Impulse coasters seems to suggest that they just run the LIMs in reverse for a very short amount of time to bring the train to a halt - but I've no confirmation of that just yet.