Further, "coroutine", like so many other programming terms, has suffered from a certain fuzziness. There's a reasonably precise term used by programming language researchers, which very few languages have. Things like Python's generators do not conform entirely to this definition, but you get people calling them coroutines anyhow, along with so many other things.
So, rather than try to justify the exact PL-research definition, let me try to explain the interesting thing about the more sloppy, broad definition. In a nutshell:
When you call a function, you pass in certain arguments. Then you start at the beginning, generating state or values in response to the input, before finally returning some more values. You then discard all the intermediate state the function generated. The idea here is that there are times when you may want to not discard the state, and/or you may want to be able to not start the function at the beginning. You may also want to pass fresh values into the middle of the function.
One of the most clear examples of where this is useful is when you create a pipeline of functions within your code. Using "functions" that can get more input as they go along, and generate output without discarding all their intermediate state, makes it very easy to compose functions together into a pipeline that only know they are in a pipeline. You can take input from a user, apply a "coroutine" that then parses their input into numbers, feed that into a "coroutine" that groups them together into a run-length delimited format, then feed that into a coroutine that writes the results to a file. Each of these routines holds on to a little bit of state, instead of discarding it. Each of these routines is "pluggable"; you can take the same pipeline, pop the "user input" bit off the front, and read from a file, or stream from the network, and the same for all the little bits. You get a composable system of functions that all know just barely enough to function (which is what makes composition easy), instead of a set of functions that knows "A ha, I'm going to get this sort of linked list, traverse it, do 'my thing', and then return a new sort of linked list", which doesn't work on network streams, or hash tables, etc. This also is an easy way to operate on a "stream" without having to have the whole stream loaded into memory, and is more cache/memory-bandwidth friendly because you're only traversing it once.
Of course, if you work at it enough, you can do this with conventional functions. But you're going to write a lot of boilerplate, you're going to get it wrong, and it probably still won't be as composable in the end. Or you'll end up manually re-implementing "coroutine"ish things yourself.
This is only one example, but it's one of the clearer ones where having things able to retain state without being forced into the strict call hierarchy that conventional function stacks require is obviously a good idea.
(Pendant note: I've tried to keep this comment very language agnostic. There's a world of difference between Haskell and Python here, for instance. I'm going for the essence of the pattern in general, not in a particular language.)