Why is anisotropic diffusion useful in image processing?
The anisotropic diffusion algorithm by Perona and Malik is the pioneering work in partial derivatives equations (PDE)-based denoising.

It applies the law of diffusion on pixel intensities to smooth textures in an image. A threshold function is used to prevent diffusion to happen across edges, and therefore it preserves edges in the image. (Unlike for instance gaussian blur filter.) This makes it very interesting if you want to remove noise, but do not want to smooth out the edges of your image, for instance if you want to use these edges to segment the image, without being perturbed by the noise.

Numerous efforts have been made that build on it, improving or extending it.

Now for where it is used, I only have a limited culture. I can cite two


Image analysis in the Life-Science field (where I work): the images that you can get out of a microscope are extremely noisy, and most of the time, it is by construction. Automated image analysis of these data often involve segmentation, for which you sometimes PDE-based algos.
Video games! For instance, try playing Mass Effect (the first one at least).