Weight is the force acting on a MASS due to it being in gravitational field. Gravitational force is the force between two objects. However the two are related.
The force of between two objects is given by
where m is conventionally the mass of a smaller object, M the mass of the larger object, r the distance between the two objects. G is is known as the gravitational constant.
If the mass m is very small compared to M then the weight of the smaller mass is the same as the Gravitational force between them.
Now the gravitational field strength of an object is given by
you can see from the equation that gravitational field strength does not depend on the mass of an object in the field.. For example the earth has gravitational field strength of about 9.81 at the surface.
The weight an object of mass m will experience is then the gravitational field strength times the mass.
W = gm
You can see by comparison then
W = m GM/r² = F
This works well for small object near a very large object however if the two objects are of similar size then then each objects gravitational field interacts with the other objects gravitational field and makes the system more complicated. But the definition of weight and Gravitational force should still hold.
*In Clarification the gravitational field strength should have the same value as the weight of an object with 1KG mass in that field.