In the paper we present an identity of the Picone type for a class of nonlinear differential operators of the second order involving an arbitrary norm $H$ in ${ℝ}^n$ which is continuously differentiable for $x\ne 0$ and such that $H^p$ is strictly convex for some $p>1$. Two important special cases are the $p$-Laplacian and the so-called pseudo $p$-Laplacian. The identity is then used to establish a variety of comparison results concerning nonlinear degenerate elliptic equations which involve such operators. We also get criteria...