TITLE: Is equality the same as identity?
QUESTION [20 upvotes]: An identity is a relation that means that whatever the number or value may be, the answer stays the same. 
But is it possible to have equality without identity?

REPLY [3 votes]: The symbol $=$ is used to denote equality of terms:
$a = b$ is read as: "$a$ is equal to $b$".
Logically, $=$ is a relation that induces an equational theory over terms. It is a reflexive, symmetric and transitive relation.
Some authors draw the following distinction:

in equations, variables are existentially quantified
in identities, variables are universally quantified.

Therefore, the key difference lies in the quantification of the occurring variables. This quantification is only clear from the context, not from the notation, because $=$ is used in both cases to denote equality.
For example, in the terminology above, the following is an identity, because it holds for all $a$:
$ a^2 = a\cdot a$
Logically, we write this as: $\forall a\, (a^2 = a\cdot a)$
And the following is an equation, because we are asking: Is there an $x$ such that $x^2$ is equal to 4?
$x^2 = 4$
Logically, we write this as: $\exists x\,( x^2 =4)$
However, other author's don't draw this distinction, and use the terms synonymously. 
Authors who do not make this distinction will call $a^2 = a\cdot a$ an equation too.