A projective transformation is a morphism of $P^n$ to $P^n$, for some $n$, determined by an $(n + 1) \times (n + 1)$ invertible matrix $A$ in the obvious way. The sets $Q$, $R$ are projectively equivalent if and only if there exists a projective transformation $f$ such that $f(Q) = R$.
I would like to know useful criteria to determine if two projective varieties are projectively equivalent.
Equivalence of projective varieties does imply that the varieties are isomorphic. But the converse does not hold. For example $\mathbb{P}^1$ is isomorphic to the conic defined by $(xy - z^2)$ in $\mathbb{P}^2$ but there is clearly no way that these are projectively equivalent because the degrees do not match. I'm not sure if that answers your question completely but I think it might be helpful to people who stumble onto this page.
Projective invariants, especially local ones, are often discussed in depth in the work of J.M. Landsberg. A particularly nice introduction constitutes chapter 3 of the book T. Ivey and J.M. Landsberg, Cartan for Beginners. You might also look at Wilcysnki, Projective Differential Geometry of Curves and Ruled Surfaces to see the local projective differential invariants of curves.
