Due to the visual properties of reflection and refraction, RGB-D cameras cannot accurately capture the depth of transparent objects, leading to incomplete depth maps. To fill in the missing points, recent studies tend to explore new visual features and design complex networks to reconstruct the depth, however, these approaches tremendously increase computation, and the correlation of different visual features remains a problem. To this end, we propose an efficient depth completion network named DistillGrasp which distillates knowledge from the teacher branch to the student branch. Specifically, in the teacher branch, we design a position correlation block (PCB) that leverages RGB images as the query and key to search for the corresponding values, guiding the model to establish correct correspondence between two features and transfer it to the transparent areas. For the student branch, we propose a consistent feature correlation module (CFCM) that retains the reliable regions of RGB images and depth maps respectively according to the consistency and adopts a CNN to capture the pairwise relationship for depth completion. To avoid the student branch only learning regional features from the teacher branch, we devise a distillation loss that not only considers the distance loss but also the object structure and edge information. Extensive experiments conducted on the ClearGrasp dataset manifest that our teacher network outperforms state-of-the-art methods in terms of accuracy and generalization, and the student network achieves competitive results with a higher speed of 48 FPS. In addition, the significant improvement in a real-world robotic grasping system illustrates the effectiveness and robustness of our proposed system.