A solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q(u)=φ(x)+ψ(t) for some single variable functions, q, φand ψ. In this paper, the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source. We obtain a complete list of canonical forms for such equations which admit the functionally separable solutions. As a result, we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source. The behavior and blow-up properties of some solutions are described.