Cloth simulation has wide applications including computer animation, garment design, and robot-assisted dressing. In this work, we present a differentiable cloth simulator whose additional gradient information facilitates cloth-related applications. Our differentiable simulator extends the state-of-the-art cloth simulator based on Projective Dynamics and with dry frictional contact governed by the Signorini-Coulomb law. We derive gradients with contact in this forward simulation framework and speed up the computation with Jacobi iteration inspired by previous differentiable simulation work. To our best knowledge, we present the first differentiable cloth simulator with the Coulomb law of friction. We demonstrate the efficacy of our simulator in various applications, including system identification, manipulation, inverse design, and a real-to-sim task. Many of our applications have not been demonstrated in previous differentiable cloth simulators. The gradient information from our simulator enables efficient gradient-based task solvers from which we observe a substantial speedup over standard gradient-free methods.