If $F=m\dfrac{dv}{dt}$ why is it incorrect to write $F\,dt=m\,dv$?

It is possible that your lecturer is telling you that, on its own, the expression $dt$ is meaningless, whereas $\int...dt$ does mean something quite specific, i.e. an operator or instruction to integrate with respect to $t$.

In contrast, $\delta t$ does mean something specific, i.e. a small increment in the value of $t$.

However, most people are fairly casual about this sort of thing.

The difference is not so much between university lecturers and highschool teachers as between mathematicians and physicists. Some mathematicians tend to frown on certain procedures that are perfectly acceptable to physicists. I was careful to write "some" because mathematicians familiar with Robinson's framework with infinitesimals do assign a perfectly rigorous meaning to formulas like $F\, dt = m\, dv$; see Keisler's beautiful textbook Elementary Calculus for details.