The ASCII table might be useful when you're dealing with encoding problems and want to know e.g. what exactly happened in a broken URL encoding. I used both of them for binary exploitation in the security class I took at university: When utilizing buffer overflows, it's important to get every byte at the right place of your payload. Sometimes, you also need to treat it as a string and that's where ASCII comes in.

These are only two specific usecases. As I said, there are a lot more. Development of binary file parsers could be another.