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.
OP wrote(v2):

> 
> *What makes an equation an 'equation of motion'?*
> 
> 
> 

As David Zaslavsky mentions in a comment, in full generality, there isn't an exact definition. Loosely speaking, [equations of motion](http://en.wikipedia.org/wiki/Equations_of_motion) are [evolution equations](http://www.encyclopediaofmath.org/index.php/Evolution_equation), with which the dynamical variables' future (and past) behavior can be determined.

However, if a theory has an [action principle](http://en.wikipedia.org/wiki/Principle_of_least_action), then there exists a precedent within the physics community, see e.g. Ref. 1. Then only the [Euler-Lagrange equations](http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation) are traditionally referred to as 'equations of motion', whether or not they are dynamical equations (i.e. contain time derivatives) or constraints (i.e. do not contain time derivatives).

References:

1. M. Henneaux and C. Teitelboim, *Quantization of Gauge Systems,* 1994.