A Callable is "A task that returns a result, while a Supplier is "a supplier of results". In other words a Callable is a way to reference a yet-unrun unit of work, while a Supplier is a way to reference a yet-unknown value.
It's possible that a Callable could do very little work and simply return a value. It's also possible a Supplier could do quite a lot of work (e.g. construct a large data structure). But generally speaking what you care about with either is their principle purpose. For example an ExecutorService works with Callables, because it's primary purpose is to execute units of work. A lazy-loaded data store would use a Supplier, because it cares about being supplied a value, without much concern about how much work that might take.
Another way of phrasing the distinction is that a Callable may have side-effects (e.g. writing to a file), while a Supplier should generally be side-effect free. The documentation doesn't explicitly mention this (since it's not a requirement), but I'd suggest thinking in those terms. If the work is idempotent use a Supplier, if not use a Callable.