Difference between findAny() and findFirst() in Java 8

No, both will not return the first element of the Stream.

From Stream.findAny() (emphasis mine):

Returns an Optional describing some element of the stream, or an empty Optional if the stream is empty.

This is a short-circuiting terminal operation.

The behavior of this operation is explicitly nondeterministic; it is free to select any element in the stream. This is to allow for maximal performance in parallel operations; the cost is that multiple invocations on the same source may not return the same result. (If a stable result is desired, use findFirst() instead.)

So to put it more simply, it may or may not choose the first element of the Stream.

With the current Oracle-specific implementation, I believe that it will return the first element in a non-parallel pipeline. However, in a parallel pipeline, it won't always, executing for example

System.out.println(IntStream.range(0, 100).parallel().findAny());

it returned OptionalInt[50] when I ran it. Anyhow, you must not rely on that.

findFirst return the first elements of the stream but findAny is free to select any element in the stream.

List<String> lst1 = Arrays.asList("Jhonny", "David", "Jack", "Duke", "Jill","Dany","Julia","Jenish","Divya");
List<String> lst2 = Arrays.asList("Jhonny", "David", "Jack", "Duke", "Jill","Dany","Julia","Jenish","Divya");

Optional<String> findFirst = lst1.parallelStream().filter(s -> s.startsWith("D")).findFirst();
Optional<String> fidnAny = lst2.parallelStream().filter(s -> s.startsWith("J")).findAny();

System.out.println(findFirst.get()); //Always print David
System.out.println(fidnAny.get()); //Print Jack/Jill/Julia :behavior of this operation is explicitly nondeterministic

What I understood is that both will return the first matched element from the stream, for example, when used in conjunction with filter?

That's not true. According to the javadoc, Stream#findAny():

Returns an Optional<T> describing some element of the stream, or an empty Optional<T> if the stream is empty. The behavior of this operation is explicitly nondeterministic; it is free to select any element in the stream. This is to allow for maximal performance in parallel operations;

while Stream.findFirst() will return an Optional<T> describing strictly the first element of the stream. The Stream class doesn't have a .findOne() method, so I suppose you meant .findFirst().