Question: Every now and then, I find myself reading papers/text talking about how *this* equation is a constraint but *that* equation is an equation of motion which satisfies *this* constraint.

For example, in the Hamiltonian formulation of Maxwell's theory, Gauss' law $\nabla\cdot\mathbf{E}=0$ is a constraint, whereas $\partial\_\mu F^{\mu\nu}=0$ is an equation of motion. But why then isn't $\partial\_\mu j^\mu=0$, the charge-conservation/continuity equation, called an equation of motion. Instead it is just a 'conservation law'.

Maybe first-order differentials aren't allowed to be equations of motion? Then what of the Dirac equation $(i\gamma^\mu\partial\_\mu-m)\psi=0$? This is a first-order differential, isn't it? Or perhaps when there is an $i$, all bets are off...

So, what counts as an equation of motion, and what doesn't?
How can I tell if I am looking at a constraint? or some conservation law?

Answer: 
An equation of motion is a (system of) equation for the basic observables of a system involving a time derivative, for which some initial-value problem is well-posed. 

Thus a continuity equation is normally not an equation of motion, though it can be part of one, if currents are basic fields.
In field theory, a conservation law just states that some quantity is conserved: if $\partial\_\mu \, \star = 0$ where $\star$ is a vector or a tensor, you can associate a conserved charge etc. - you know the spiel I guess.

Constraints are something you impose by hand (or by experiment). 

Finally, equations of motion are dynamical equations that follow from the Euler-Lagrange equation. Both the Dirac equation and $\partial^\mu F\_{\mu \nu} = 0$ satisfy that criterion. [However, if you choose a gauge, it becomes much more clear that you're dealing with a PDE for the field $A\_\mu$, such as $\square A\_\mu = 0.$] Also note that both of them involve $\partial\_t$ or $\partial\_t^2$. Note that $\nabla \cdot \mathbf{E}$ only involves spatial derivatives, so it gives no dynamics.

In your example, $\partial^\mu j\_\mu = 0$ is a classical conservation law that doesn't describe how some microscopic quantity evolves in time - you don't derive it from Euler-Lagrange.