Is it possible for two things to be equal, but not identical? For example, would it be correct to say that 2+2 is equal to 1+3, but 2+2 is not identical to 1+3? If not in that particular case, is there at least one case where two objects are equal but not identical? Or is equality and identity the same relation?
This comes down to the way we can use "equal" to mean "identical" (the same in all ways) or "equivalent" (same in a relevant way).
This makes sense in @Steven Saban's example of 4 quarters being equivalent to a 1 dollar bill in the relevant way of currency value (but not in all ways, such as physical appearance, so they are not identical).
So, considering the question of a "case where two objects are equal but not identical" this is not just possible but common. Obviously if we interpret "equal" to mean "identical" this trivially won't make sense, but where "equal" means equivalent, such as where "4 quarters is equivalent to 1 dollar bill" in the context of currency value but not appearance or "perro is equivalent to dog" in the context of semantics but not spelling, are cases of this.
Note: In mathematics the usages of "equal" and "equivalent" have their own important differences, though it is shifted more to a utility of mathematics than our everyday language so should be considered within that field.
2 + 3 is equal to 5 but not identical to it.
Equal means "the same in one or several particular respects", respect which are usually left implicit when understood from the context.
Identical means "the same in all respects".
So 2 + 3 = 5 is true because 2 + 3 and 5 have the same value. And it goes without saying that value is the respect in which they are equal.
So they are the same in respect of their value, although they are obviously different as indicated by their forms and so are not identical: 2 + 3 is an operation and 5 is a number.
2 + 3 and 5 have identical values since their values are the same in all respects. Here we have to narrow down the scope of identity by being explicit it is their values which are the same: identical values. This because Identical on its own has maximal scope, beyond just value.
Equality doesn't require an exact likeness in all respects:
Merriam Webster Equal:
"Equal 1 of 3 adjective
1a (1): of the same measure, quantity, amount, or number as another
(2): identical in mathematical value or logical denotation : EQUIVALENT
b: like in quality, nature, or status
c: like for each member of a group, class, or society
provide equal employment opportunities 2: regarding or affecting all objects in the same way : IMPARTIAL
3: free from extremes: such as
a: tranquil in mind or mood
b: not showing variation in appearance, structure, or proportion
4a: capable of meeting the requirements of a situation or a task
b: SUITABLE bored with work not equal to his abilities
equal 2 of 3 noun
1: one that is equal insists that women can be absolute equals with men - Anne Bernays
2: an equal quantity
equal 3 of 3 verb equaled or equalled; equaling or equalling
1: to be equal to especially : to be identical in value to
2 archaic : EQUALIZE
3: to make or produce something equal to
If two properties are the same then those are equal, but the objects are not necessarily identical.