Is there ever a situation where the distinction between $F = m \frac{dv}{dt}$ and $F = \frac{dp}{dt}$ is important? I can't think of a situation where one is true and not the other (assuming only conservation of momentum).
Edit: Obviously it is important to take a changing mass into account (e.g. for a rocket) when you're considering a full time evolution, i.e. $F(t) = m(t) \frac{dv}{dt}$ (or in the relativistic case, perhaps something like $F(t) = m(t) \frac{d}{dt} \left( \frac{p}{m} \right)$ with $m$ the rest-mass). And perhaps there is a nontrivial relationship between the rate of change of mass, and the forces being exerted (again, e.g. with a rocket --- where the mass loss is tied to the propulsion). What is not clear is that there should ever be a $F = v\frac{dm}{dt}$ term.
Edit 2: My understanding of the solution:
There should not be a $dm/dt$ term, as pointed out by @garyp. The change in momentum expression is, however, more accurate because $p \neq mv$ in general (e.g. in relativistic cases, or when considering massless systems). It would seem that either one must take the caveat that $dp/dt$ cannot be used for mass-varying systems, or take the much less conceptual or aesthetically pleasing expression that $F = m \frac{d (\gamma v)}{dt}$ (which still only applies to classical systems).
