Computation is the difference. In Lean, applying the universal property of the quotient (`Quotient.lift f Hf`) to an element that is of the form `Quotient.mk a` reduces to `f a`.
This rule is fine in itself, but the Lean developers were not sufficiently careful and allowed it to apply for quotients of propositions, where it interferes with the computation rules for proof irrelevance and ends up breaking subject reduction (SR is deeply linked to computation when you have dependent types!) [0]. It is not really a problem in practice though, since there is no point in quotienting a proposition.
This rule is fine in itself, but the Lean developers were not sufficiently careful and allowed it to apply for quotients of propositions, where it interferes with the computation rules for proof irrelevance and ends up breaking subject reduction (SR is deeply linked to computation when you have dependent types!) [0]. It is not really a problem in practice though, since there is no point in quotienting a proposition.
[0] see the end of section 3.1 in https://github.com/digama0/lean-type-theory/releases/downloa...