The distinction between a ‘binary relation’ and ‘an ordered pair’ is particularly germane when considering the set membership relation, ‘$\in$’, because it clearly illustrates a case of the chicken vs. the egg.
As the Wikipedia article states, a set theory is some logic (say first order with identity for arguments sake) with the addition of the $\in$-relation and axioms dictating its use.
Now while not necessary, ordered pairs are often defined in terms of a set theory as a later development. When defined in such a way, an ordered pair is essentially reduced to a logical statement involving the $\in$-relation, as are the sets which aggregate the ordered pairs.
That last bit is important, for while we can certainly describe ‘binary relations’ in terms of sets of ‘ordered pairs’, consider what happens when you try to define the $\in$-relation in terms of such set theoretical ‘ordered pairs’. Without some additional machinery our definitions turn circular: ordered pairs in terms of the $\in$-relation, and the $\in$-relation in terms of (sets of) ordered pairs.
The take away here, I believe, is that while it can be useful to talk about relations in general in terms of sets of ordered sequences, particularly when relations are the object of study, in practice relations are properly a part of the underlying logical language used to make assertions about objects, not objects about which we make assertions.