First of all we are all about 4000 miles from Earth because (as shown by Newton) the force is calculated from the center of all objects.  This means however, that an object weighed at the poles would be slightly heavier than an object weighed at the equator (with appropriate variations between the poles and the equator).  The reason is, because of the equatorial; bulge, there is less distance to the center of Earth at the poles and so objects there are closer to Earth than those at the equator and consequently weigh more.

Another factor adds to the difference in weight of objects at the poles compared to the equator.  That is centrifugal force.  Every place on earth makes a complete circuit every day.  However the speed increases from zero at the poles to about 1000 mph at the equator.  Since it is a curved path that means the centrifugal force (in this case the force tending to throw things into space), tends to lighten weights at the equator a little more.

We can also compare weights of something passing over a cave filled with gold (Fort Knox) with the same thing passing over an empty cave.  The added mass of the gold should increase the weight.  Ore deposits (of heavy metals) should have the same effect.  We can compare climbing a mountain with staying at the bottom.  Climbing a mountain puts one further from earth and thereby should decrease weight.  In effect then, anything that puts one farther from Earth or closer to it or puts something heavier or lighter under you causes a change (slight because the changes are slight) in weight.

There are some consequences for this variation.  For example, in a fjord, the water on one side would be attracted to the rocks of that side and the water on the other side would be similarly and oppositely attracted to the rocks of the other side.  More generally, water (the easy definer of a horizontal surface) would be attracted to anything heavier.  This means the oceans would curve up to the near shore.  This complicated surface is actually the true horizontal and is called the geoid.  Actually, the geoid is so complex that geophysicists can't manage it mathematically so they average the geoid into something mathematically manageable called the spheroid.  So, rather than having a spherical Earth, we actually have an Earth that is an oblate spheroid of revolution.

ISOSTASY
Isostasy is the idea the light rocks (especially continents and their mountains, but also ocean basin rocks) float on denser materials below.  Flotation means that the object floating sinks down until it displaces exactly its weight in material below.  How could this have been discovered in the mid-19th century by Pratt (and then with a modification by Airy)?

This story starts after Britain conquered India and formed the Raj.  George Everest (for whom the mountain was named) was given the job of mapping.  He noticed that mapping by triangulation (a standard land mapping technique) led to different distances from Central India to the vicinity of the Himalayas (Kaliana to Kalianpur).  These differences in distances, about 500 feet, were far too great to be accounted by any errors typical of mapping (or any other measuring -- all measuring has errors to it).

The inconsistencies were presented to John Pratt (an English clergyman and mathematician who spent many years as Archdeacon of Calcutta).  Pratt reasoned that since the mapping went from some distance from the Himalayas to close to them that the plumb bob used for determining vertical was being attracted more toward the Himalayas at Kalianpur than at Kaliana.   This would reduce the apparent distance because it would make it appear that Kalianpur was further south than it really was.

With that first thought he then studied the situation (estimated the mass of the Himalayas etc.).  Much to his surprise he found that not only did the Himalayas explain the error, but also the error should have been more than three times larger (more than 1500 feet).  This meant that mountains do not have as much material as they appear to have.  One mathematical (mathematicians do not -- if they are any good -- limit themselves to reality) solution would be that the mountains were hollow. This would solve the problem mathematically, but all those streams have been cutting through the Himalayas without revealing any caverns so, in reality, despite its mathematical attraction, hollowness cannot be the answer.

Pratt came to the thought, probably because he knew oceanic are rocks were denser than continental rocks and because of the mathematical fit, that areas of different heights were made of different densities of rocks floating (when formed) on even denser materials below.  Airy thought the rocks of the world were similar and so his solution was to put deeper roots below higher places.

In actuality, a combination of the two models is closer to the truth - we know that different rocks do have different densities (Pratt model), but that beneath continents there are tectonic "roots" (Airy model).