(I'm not completely happy with the answers so far, so here is my attempt...)
I think that Kevin Wright's comment ("You can make a Promise and it's up to you to keep it. When someone else makes you a promise you must wait to see if they honour it in the Future") summarizes it pretty well, but some explanation can be useful.
Futures and promises are pretty similar concepts, the difference is that a future is a read-only container for a result that does not yet exist, while a promise can be written (normally only once). The Java 8 CompletableFuture and the Guava SettableFuture can be thought of as promises, because their value can be set ("completed"), but they also implement the Future interface, therefore there is no difference for the client.
The result of the future will be set by "someone else" - by the result of an asynchronous computation. Note how FutureTask - a classic future - must be initialized with a Callable or Runnable, there is no no-argument constructor, and both Future and FutureTask are read-only from the outside (the set methods of FutureTask are protected). The value will be set to the result of the computation from the inside.
On the other hand, the result of a promise can be set by "you" (or in fact by anybody) anytime because it has a public setter method. Both CompletableFuture and SettableFuture can be created without any task, and their value can be set at any time. You send a promise to the client code, and fulfill it later as you wish.
Note that CompletableFuture is not a "pure" promise, it can be initialized with a task just like FutureTask, and its most useful feature is the unrelated chaining of processing steps.
Also note that a promise does not have to be a subtype of future and it does not have to be the same object. In Scala a Future object is created by an asynchronous computation or by a different Promise object. In C++ the situation is similar: the promise object is used by the producer and the future object by the consumer. The advantage of this separation is that the client cannot set the value of the future.
Both Spring and EJB 3.1 have an AsyncResult class, which is similar to the Scala/C++ promises. AsyncResult does implement Future but this is not the real future: asynchronous methods in Spring/EJB return a different, read-only Future object through some background magic, and this second "real" future can be used by the client to access the result.
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