Does axiomatizability in zeroth-order logic have important consequences?

Does axiomatizability in zeroth-order logic have important consequences?

If a theory is equationally axiomatizable, this has important consequences
(that are studied e.g. in universal algebra).
However, many theories fail to be equationally axiomatizable - examples
include fields, integral domains, and partially ordered sets. Nonetheless,
all of these examples are zeroth-order axiomatizable. Does this have
important consequences?



For instance, the theory of partially ordered sets is generated by the
following axioms.
$x \leq x$
$(x \leq y) \wedge (y \leq x) \rightarrow x = y$
$(x \leq y) \wedge (y \leq z) \rightarrow x \leq z$