Viscosity solutions are the "appropriate" notion of solutions for second-order elliptic equations in nondivergence form, and for some classes of first-order equations. Here is a brief summary of why.
From the point of view of applications, the viscosity solution is almost always the right solution. For example, in optimal control theory, it has long been known that if the value function is smooth, it satisfies a certain PDE-- but the value is known to not be smooth in most cases. When Crandall and Lions invented viscosity solutions, it was clear immediately that the viscosity solution is precisely the value function. This story has played out over and over again, for many different applications, to the point that people in the field are completely shocked and stunned when there is some reason to consider a notion of solution other than viscosity solution.
From a mathematical point of view, viscosity solutions are natural. For equations in nondivergence form, energy methods are unavailable. Therefore, all one usually has is the maximum principle. Viscosity solutions are to weak solutions as the maximum principle is to energy methods. The term "viscosity solution" is rather unfortunate in this sense-- they should be called "comparison solutions" or something. The point is, these equations should satisfy a maximum principle. So it makes sense to define your weak solution as a one for which the maximum principle holds when you compare to smooth functions.
Finally, in the cases where there is overlap, like for the p-Laplacian, viscosity solutions are equivalent to (bounded) weak solutions in the integration-by-parts sense.