What is the difference between chemical equilibrium and dynamic equilibrium?
Chemical equilibrium is a type of dynamic equilibrium, but not every dynamic equilibrium is a chemical equilibrium.

In a chemical equilibrium there is no change on the macroscopic scale. That means that if you look at the system it seems like nothing is happening, but at molecular scale there are reactions going on and the rate of forward reaction = rate of backward reaction.

There is another example of dynamic equilibrium (although not the best one, as it is not a closed system):
Imagine a bath into which water is flowing in at constant rate. Water is also flowing out of the bath at the bottom with the same rate. 
This is an example of dynamic equilibrium, which has nothing to do with chemistry. If you look at the bath, the water level is constant, so on the macro level nothing is happening, but there is flow of water, so on micro level water from the tap is replacing water flowing out of the bath.