Digital image processing is an extension of digital signal processing and linear system theory into two dimensional signals.
Image processing involves all low level tasks such as filter design and filtering, spatial scaling, sampling, intensity manipulations, geometry manipulations, Fourier analysis and spectrum analysis, motion estimation, noise reduction, edge detection, image enhancement and restorations etc.
Computer vision involves high levels tasks such as object recognition, scene recognition and image understanding; a more computer science perspective than a DSP type. Note that computer vision inevitably will use image processing at its preprocessing or post processing steps.