Well, as you say, these two categories are isomorphic, so it's going to be hard to say how they differ! They only differ in the names the maps are given.

Maybe it would help to recap the definitions. I'll take them from p.39-40 of Peter Johnstone's book *Stone Spaces*.

A **frame** is a complete lattice $A$ satisfying the infinite distributive law
$$
a \wedge \bigvee S = \bigvee \{ a \wedge s | s \in S \}
$$
($a \in A, S \subseteq A$). A **homomorphism of frames** is a function preserving finite meets and arbitrary joins. This defines the category **Frm** of frames.

Note (as you did) that every homomorphism of frames has a right adjoint.

The category **Loc** of **locales** is the opposite of the category of frames. Morphisms in **Loc** are called **continuous maps**.

Then Johnstone says: "We adopt the convention that if $f: A \to B$ is a continuous map of locales, we shall write $f^*: B \to A$ for the corresponding frame homomorphism, and $f_*: A \to B$ for the right adjoint of $f^*$."

So in Johnstone's convention (which is the one I know), the elements of **Loc**$(A, B)$ are identified with frame homomorphisms $B \to A$. In Borceux's convention, the elements of **Loc**$(A, B)$ are identified with order-preserving maps $A \to B$ that are right adjoint to frame homomorphisms.

I guess the other thing to say is that when you're dealing with ordered sets, adjoints are *genuinely* unique (not just unique up to isomorphism). So taking the right adjoint of a frame homomorphism is a bijective process.

I don't know what else to say. It's really just a matter of naming.