Based on the advantages of nonconvex variational models on image edge-preserving and contrast-preserving, this paper introduces a new nonconvex and nonsmooth variational model together with a fast algorithm for the Poisson noise removal. The proposed model consists of a regularization term and a data fidelity term. The regularization term is formulated by a nonconvex Lipschitz potential function composed of the first-order derivative of images, while the data fitting term is depicted by the nonlinear Kullback-Leibler divergence. By using the proximal linearization strategy, the proposed nonconvex and nonsmooth model can be converted into a series of convex models, which are able to be solved by alternating direction method of multipliers. Moreover, we can also prove the monotonic decreasing property of the objective function value sequence. Numerical experiments show that our model with the proposed algorithm is effective for eliminating Poisson noise and obtains higher SNR values compared to classical methods.