An object is defined by an OBJECT_ID, which is unique. If A and B are objects and
A == B is true, then they are the very same object, they have the same data and methods, but, this is also true:
A.OBJECT_ID == B.OBJECT_ID
if
A.Equals(B) is true, that means that the two objects are in the same state, but this doesn't mean that A is the very same as B.
Strings are objects.
Note that the == and Equals operators are reflexive, simetric, tranzitive, so they are equivalentic relations (to use relational algebraic terms)
What this means:
If A, B and C are objects, then:
(1) A == A is always true; A.Equals(A) is always true (reflexivity)
(2) if A == B then B == A; If A.Equals(B) then B.Equals(A) (simetry)
(3) if A == B and B == C, then A == C; if A.Equals(B) and B.Equals(C) then A.Equals(C) (tranzitivity)
Also, you can note that this is also true:
(A == B) => (A.Equals(B)), but the inverse is not true.
A B =>
0 0 1
0 1 1
1 0 0
1 1 1
Example of real life:
Two Hamburgers of the same type have the same properties: they are objects of the Hamburger class, their properties are exactly the same, but they are different entities. If you buy these two Hamburgers and eat one, the other one won't be eaten. So, the difference between Equals and ==:
You have hamburger1 and hamburger2. They are exactly in the same state (the same weight, the same temperature, the same taste), so hamburger1.Equals(hamburger2) is true. But hamburger1 == hamburger2 is false, because if the state of hamburger1 changes, the state of hamburger2 not necessarily change and vice versa.
If you and a friend get a Hamburger, which is yours and his in the same time, then you must decide to split the Hamburger into two parts, because you.getHamburger() == friend.getHamburger() is true and if this happens: friend.eatHamburger(), then your Hamburger will be eaten too.
I could write other nuances about Equals and ==, but I'm getting hungry, so I have to go.
Best regards,
Lajos Arpad.