What is the difference between thenApply and thenApplyAsync of Java CompletableFuture?

The difference has to do with the Executor that is responsible for running the code. Each operator on CompletableFuture generally has 3 versions.

1. thenApply(fn) - runs fn on a thread defined by the CompleteableFuture on which it is called, so you generally cannot know where this will be executed. It might immediately execute if the result is already available.
2. thenApplyAsync(fn) - runs fn on a environment-defined executor regardless of circumstances. For CompletableFuture this will generally be ForkJoinPool.commonPool().
3. thenApplyAsync(fn,exec) - runs fn on exec.

In the end the result is the same, but the scheduling behavior depends on the choice of method.

You're misunderstanding the examples you quoted. In both examples, the second function has to wait for the first function to complete. Whenever you call a.then___(b -> ...), input b is the result of a and has to wait for a to complete, regardless of whether you use Async or not.

The actual example in the article is

CompletableFuture<String> receiver = CompletableFuture.supplyAsync(this::findReceiver);

receiver.thenApplyAsync(this::sendMsg);  
receiver.thenApplyAsync(this::sendMsg);  

Notice the thenApplyAsync both applied on receiver, not chained in the same statement. This means both function can start once receiver completes, in an unspecified order. (Any assumption of order is implementation dependent.)

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More technical explanation

I must point out that thenApply and thenApplyAsync are terribly named and are confusing to the unfamiliar. There is nothing in thenApplyAsync that is more asynchronous than thenApply from the contract of these methods.

The difference between the two has to do with on which thread the function is run. The function supplied to thenApply may run on any of the threads that

1. call complete
2. call thenApply on the same instance

while thenApplyAsync either uses a default Executor (a.k.a. thread pool), or a supplied Executor.

Asynchrony != threads

thenApply/thenApplyAsync, and their counterparts thenCompose/thenComposeAsync, handle/handleAsync, thenAccept/thenAcceptAsync, are all asynchronous! The asynchronous nature of these function has to do with the fact that an asynchronous operation eventually calls complete or completeExceptionally. The idea came from Javascript, which is indeed asynchronous but isn't multi-threaded.