In a linear supply, the output voltage is maintained by an electronically variable resistor. Usually a power MOSFET.
Pros: - simple design, cheap, little electrical noise.
Cons: - lots of waste heat, lots of waste heat, can be slow to respond to transients, usually larger than switchers for the same power rating.
In a switching power supply the output voltage is controlled by varying the percentage of time the controlling switch is on.
Pros: - usually much smaller than a linear of the same power rating, can have very good transient response.
Cons: - more complicated to design and get to work as desired, lots of electrical noise that should be dealt with.
That is a quick and simple answer.