It seems to me that (classical) computable mathematics and constructive mathematics follow roughly the same program, i.e. only working with objects which can explicitly be constructed (by an algorithm, etc.). A key difference, of course, is that constructivism rejects the law of excluded middle over infinite sets, while classical computability (presumably) makes full use of it.
So, would a fair characterisation of the two be that $$\mathsf{computability}\ =\ \mathsf{constructivism}\ +\ \mathsf{law\ of\ excluded\ middle}$$ Are there any other key differences between these two ways of thinking?