- Density is a property of all states of matter (solids, liquids and gases) but viscosity is a property which only fluid have (especially liquids). The reason why only fluids have viscosity is that force viscous force to act they must be some flow (or) to say only some flow causes viscous force. Since there is no such flow in solids so there is no topic of viscosity in solids. Also, in gases this is very negligible which is discussed in the next point.

- They are quite different on how they are defined. Density is defined as the mass per unit volume whereas viscosity is defined as the resistive force acting on a body which is freely falling inside a liquid. But don't confuse viscosity with buoyant force. Buoyancy is acting because of the pressure acting on the submerged part of the body, from the liquid and viscosity is caused due to the friction between liquid molecules and the surface of the body, caused mainly due to the movement of the body inside the liquid. In gases the molecules are placed so apart such that the individual friction between a molecule and the body can't resist the motion but in liquids the molecules are little closely packed which makes the total friction from all molecules enough to resist the motion of body. This is the reason why viscosity is dominant in liquids only. 

- Density is such a property which describes the liquid itself but viscosity describes the motion of a body present in it.