Recursive equations vs. inference rules
Inference rules are "more general". A recursive equation, actually any equation $E_1 = E_2$ is like an inference rule without premisses:
$$\frac{}{E_1 = E_2}$$
A general inference rule may have premisses, and so it is like a conditional equation.
Only very special kinds of inference rules may be equivalently expressed as equations. For instance, it seems difficult to express
$$\frac{a \bullet c = b \bullet c}{a = b}$$
and
$$\frac{add(x,add(y,z)) = add(x,z)}{add(y,z) = z}$$
as equations.