TITLE: Differences between logic with and without equality
QUESTION [10 upvotes]: By logic without equality I mean those kind of logics where equality is treated as a binary relation satisfying some axioms, as opposed to a logics where equality is a logical symbol satisfying some inference rules.
In my understanding there should be no difference at the syntactic level between logics with or without equality since the syntax should not be able to distinguish a logic-primitive predicate or any other predicate (but in case I am wrong please correct me).
So I am wondering what are the differences at a semantic level between these kind of logics.
In particular I am wondering whether all the definitions and theorems of classical model theory can be transported verbatim to the without-equality logics.
If it is possible I would also appreciate references that treat the subject.
Addendum: After Andrej Bauer's answer I realize it would be better to add some specification.
I am interested basically in logical theories where every theory comes equipped with an axiom-schema for substitution, that is a family of axioms of the form 
$$\forall x,y. (x=y) \land \varphi(x) \rightarrow \varphi(y)$$
where $\varphi$ is a formula of the language.
In short I am curious to know what changes we get if we change the semantics of the symbol of equality.
Addendum 2: also it seems from the answer provided so far that the model theory should remain unchanged but it does not look like it to me.
First of all while quotienting for the equivalence relation should provide a (elementary?) equivalent model it does not (necessarily) preserves the homomorphisms: consider a set with an equivalence relation that identifies all elements, the corresponding quotient structure should be the terminal model (in the sense of category theory) but the first model clearly does not have to be a terminal object in the category.
Also it seems to me that one could lose the characterisation of homomorphisms as mapping that preserve atomic formulas, since, if we drop the requirement of interpreting the equality symbol as the identity relation, then we could have mappings that preserve atomic formulas but for some operations symbols do not respect the external equality: i.e. we could have a mapping $f \colon A \to B$ and an operation symbol $o$ such that $$B \models f(o^A(a)) = o^B(f(a))$$
but $f(o^A(a))$ and $o^B(f(a))$ are not identical.
It seems rather difficult to find references that deal with this kind of phenomena, that is why I would really appreciate if someone could provide some pointers. 
Thanks in advance.

REPLY [4 votes]: When dealing with model theory without equality, I think it is better not to deal with isomorphisms (functions) but with isomorphism relations. For example, in order to recover Fraïssé's theorem, one may consider partial isomorphism relations because one cannot infer the existence of a partial isomorphism from the satisfaction of the same atomic formulas when equality is absent. 
What is gained in doing this? I think that something is gained, and it is not extra generality. I will try to make a point:
1) A natural question in foundational investigations is: what are the minimal resources that a classical logical system must have in order to be a possible vehicle for classical mathematical reasoning? One can, with first-order logic without equality, axiomatize set theory with only one primitive binary relation (membership). So, equality is certainly not part of the minimum. Is this (first-order logic with membership but without equality) the minimal vehicle? For one interpretation of what is meant by classical mathematical reasoning, one can prove, using some model theory without equality, that it is not.
2) For pedagogical purposes, results in logic without equality are usually simpler to prove, one can get more focused on the essential ideas, not being distracted by some technical details. For example, the construction of the canonical structure is simpler, either from Hintikka sets or from maximal Henkin sets. Also, proof theory without equality is simpler. For example, one need not talk about quasi-tautologies in Herbrand's theorem without equality. The proof-theoretic characterizations of logical theorems are clearer in this setting. It is more general and simpler.    
The following paper of mine deals with some model-theoretic stuff: 
https://link.springer.com/article/10.1007/s11787-015-0126-8