In first order logic, how do we normally represent a statement?
This is a good question and comes down to what you want to achieve with the Mother entity.

In your first example, Mother(x, y) is a (binary) relation.  That means that you can have as many ys as you would like per x.  With Mother as a relation, you can also make formula that checks whether you have two ys that satisfies motherhood for x.

In your other example, hasMother(x) is a predicate that can only be true or false.  How many mothers do x have? Impossible to say.  Who is x's mother?  We can't say.

You could also conceive of a mother(x) → y function that maps an x to its mother y, but then you can have only one mother, and every x needs to have a mother (in the model).

It all comes down to (a) what is a natural representation of "the world" (whatever you want to model), and (b) what you want to achieve (whatever you want to get out of your model).