Is there a difference between Weight and Force of Gravity?

Understanding this question already has an accepted answer, I was confused by this as well and did some additional research that I will share.

The first is the definition of weight. I first understood the definition of weight from the textbook I was reading as the gravitational force that exerts on a body. From this definition, there is no difference between ${F_g}$, the gravitational force, and $\vec w$, weight force.

That said, after re-reading and searching all the internet sufficiently, the definition more accurately used for weight is the gravitational force that the earth exerts on the body or "If you are on another planet, your weight is the gravitational force that planet exerts on you." Young, H. and Freedman, R. (2016). University Physics with Modern Physics (14th ed). Pearson Education, Inc, pp. 114.

With that in mind, typically weight force is a particular type of the gravitational force that refers to the gravitational force on a body (e.g. on the moon, on Mars, on Earth). If you are referring within space such as the gravitational force of Earth on the International Space Station (ISS) or an astronaut on the ISS, they are in constant free-fall, and for sake of clarity, the force causing that free-fall is just called gravitational force since there is the sensation of weightlessness. It would be a misnomer to say how much does an astronaut weigh on the ISS and more accurate to say what is the gravitational force the earth (or the ISS) exerts on the astronaut.

The same can be said of celestial bodies, such as the moon. One would not say, how much does the moon weigh. One would say, what is the gravitational force the earth has on the moon (and so forth for the Earth to the Sun and Mars to the Sun).

When being asked for the weight force, it is generally implicitly understood, it's the gravitational force of an object on the earth (or it will be specifically called out on what body and the acceleration due to gravity on that body, e.g. find the weight force of a person of 68 kg mass on the moon which has an acceleration due to gravity of $1.620 \dfrac {m}{s^2}$). When asked for the gravitational force, the 2 object's mass will be provided or the needed data to calculate their mass.

Here is a comparison at the formula/math definitions.

Weight Force: $$\vec w = m\vec g $$ where m is the mass of the object and $\vec g$ is the acceleration vector due to gravity (e.g. has both magnitude and direction).

Gravitational Force between 2 objects:

$${F_g} = G * \frac {m_1 m_2}{r^2}$$ where G is the gravitational constant $$\frac {6.67384(80) * 10^{-11} m^3}{kg * s^2}$$ and $m_1$ is the mass of object 1 and $m_2$ is the mass object 2 with $r$ being the distance between the 2 objects.

If we were to calculate both values for 150 lb person at sea level, we would find that both equal $667 N$ (rounding to 3 significant figures). To simplify the calculation of gravitational force on the Earth, Newton derived the weight force relationship.

Hope this helps.

While both are forces, weight is generally specific to any sum of forces you feel reciprocated as a normal force (or tension). I can feel heavy in a centrifuge because of the centrifugal force. When I find myself sitting in my chair (safely away from centrifuges) the astronauts on the International Space Station and I are both are subject to the force of gravity--but only I will feel my weight in my chair as the chair pushes back up against me. The astronauts are in free fall and do not feel the sensation of weight ignoring tidal effects.

Weight is a subtle concept, because we are so used to it we don't even notice it anymore. You'll find surprising how badly some students grasp the concept of weight, despite the fact they are firmly sitting on a chair in the very same moment.

Anyway, weight is the force an object experiences when inside a gravitational field. That means me, you, StackExchange servers, air, but even ISS, which is orbiting apparently "weightless", experience weight. You can be tricked to feel less or more heavy by applying other forces, for example in a centrifuge, on a roller-coaster, or when accelerating inside a car.

Even when you are free-falling you are experiencing weight, in fact it's exactly the reason you're falling down.

But weight is not really gravity, because sometimes it doesn't really make sense to speak about weight, even in a gravitational field.

First, whereas you can have gravity without weight (e.g. light is bended by gravitational fields, altough it has not weight, because it is massless), you cannot have weight without gravity.

Second, gravity acts both ways: as Earth is pulling you down on the ground, you are pulling Earth "up" with the same strength. Thus by talking only about your weight you're being a little selfish and hurting Earth's feelings, as well as Newton's third law's.