This is more of a physics explanation, but here we go.
Mass is a measure of how much "matter" is in an object. Weight is the force applied onto an object by gravity. Weight itself can be related to mass like this:
where g is a gravitational constant. For our purposes, it's defined by whatever planet you are on. Following this, we can demonstrate that mass is NOT the same thing as weight if we take two objects of the same mass and put them on different planets.
Let E refer to Earth and F refer to Mars
Following this, we can see clearly that weight is not the same as mass:
If weight was the same thing as mass, the two values would be the same, as the mass of the two objects is the same. But since weight is defined in the context of gravity, they are not.