Robot localization using a built map is essential for a variety of tasks including accurate navigation and mobile manipulation. A popular approach to robot localization is based on image-to-point cloud registration, which combines illumination-invariant LiDAR-based mapping with economical image-based localization. However, the recent works for image-to-point cloud registration either divide the registration into separate modules or project the point cloud to the depth image to register the RGB and depth images. In this paper, we present I2PNet, a novel end-to-end 2D-3D registration network, which directly registers the raw 3D point cloud with the 2D RGB image using differential modules with a united target. The 2D-3D cost volume module for differential 2D-3D association is proposed to bridge feature extraction and pose regression. The soft point-to-pixel correspondence is implicitly constructed on the intrinsic-independent normalized plane in the 2D-3D cost volume module. Moreover, we introduce an outlier mask prediction module to filter the outliers in the 2D-3D association before pose regression. Furthermore, we propose the coarse-to-fine 2D-3D registration architecture to increase localization accuracy. Extensive localization experiments are conducted on the KITTI, nuScenes, M2DGR, Argoverse, Waymo, and Lyft5 datasets. The results demonstrate that I2PNet outperforms the state-of-the-art by a large margin and has a higher efficiency than the previous works. Moreover, we extend the application of I2PNet to the camera-LiDAR online calibration and demonstrate that I2PNet outperforms recent approaches on the online calibration task. Source codes are released at https://github.com/IRMVLab/I2PNet.