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A crankshaft has one (or more) offset sections where a connecting rod is attached around it. The connecting rod moves back-and-forth (or up-and-down) ONCE for every rotation of the crankshaft. The crank will always move in a circle, moving the connecting rod in a smooth sine wave like motion. The power can flow in either direction. That is, the crankshaft can drive the connecting rod, or the connecting rod can drive the crankshaft.
A camshaft has a cam follower that rides on the cam. The cam on the camshaft can have more than one lobe (high spot) on it, so it is possible for the cam follower to move back-and-forth (or up-and-down) more than once for each rotation of the camshaft. The cam can have a complex shape, so the cam follower can move in motion patterns that are non-sine wave like. Usually the power flows in only one way. The camshaft will drive the cam follower, but the cam follower usually cannot drive the cam.
If a significant amount of power needs to be transmitted, a crankshaft is usually used. If motion (with minimal power) needs to be transmitted, particularly if it is a non sinusoidal motion, then a camshaft is used.
In a typical piston engine (gasoline, diesel, etc) uses a crankshaft to convert the linear motion of the pistons into rotational motion. The crankshaft is good here because it must be able to push the pistons as well as the pistons pushing it. Also, a substantial amount of power must be transmitted.
In the same engine, a camshaft is used to operate the valves. The valves in the engine should stay fully closed most of the time, then open quickly, stay open for a bit, and then close quickly. The camshaft works well to produce this non-sinusoidal motion where little power needs to be transmitted.
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