“Shatter” implies that you broke something into many small pieces, like glass or ceramics as Adriel said. “Smash” could also mean this, but not necessarily. “Smash” could also mean more generally that you break something with a lot of force. Think of “Hulk smash!”

As mentioned below, shatter is the action of something breaking into small pieces, or destroyed (i.e. "My heart was shattered when she left me;" "The hopes of my getting that job were shattered when they said I needed to be fluent in 5 languages." Smash, on the other hand, is the action of something being pressed down upon

Like the others said, "shatter" usually means that something breaks into many very tiny pieces. A car can be smashed (in a car accident, for example), but only the car windows will shatter. You can smash a TV by throwing it on the ground, but only the screen will shatter. (Other parts of the TV will be broken, but only the screen will break into tiny fragments that scatter everywhere.)

They are practically the same but "shatter" tends to be used more often when describing the breaking of glass or pottery.