I am a math student and I am taking my first physics course of Mechanics. But I don't know the difference between the $w=mg$ force and $F=G\cfrac{m_1 m_2}{r^2}$ (why we don't use the first one when talking about the interaction between our planet and me).
For me, the two forces are related since $g$ is the gravitational acceleration. I know that $F=G\cfrac{m_1 m_2}{r^2}$ is the force with which the earth attracts us (because we have mass).