Here are several related uses.

Inline uses hashing to know when it has to recompile and relink libraries, versus when it has already done so and can use the old version. This means that you only pay for compilation once.

Rsync uses it to identify when and where files have changed without having to send the whole file across the wire. Instead it sends md5 hashes back and forth, and when they differ it starts sending hashes of part of the file doing a binary search to generate a diff which it can then use to transfer a patch with which it updates the file.

In a website you might use it to generate a handle representing a complex data structure so that you can store the user information locally and just pass a small amount of information back and forth to the client. An example of why you would want to do this is when you have a gif whose display requires a large amount of form data to put together. You can't pass all of the information in the URL for the gif. You can pass the appropriate MD5 hash.