Do differential forms and ordinary differential equations have identical solutions?

Question

Consider the 1-form $$ M(x,y)dx + N(x,y)dy = 0 $$ and the ODE $$ M(x,y) + N(x,y)y'=0. $$ These functions seem to be identical, and if one were to abuse notation, the first equation, multiplied by $\frac{1}{dx}$ (whatever that means) is equivalent to the second. I believe that they have identical solutions, but my professor said that their solutions are not identical; namely that the solutions to the first are level curves of some function, can be implicit, and are expressed in the form $f(x,y) = c$, where c is constant, while the second has solutions that are functions, explicit, and are expressed in the form $y(x) = g(x,c)$, where c is constant.