Does gravity differ from place to place on Earth
There is an entire field of Geophysics called gravimetry dedicated to measuring the magnitude of the gravitational field. 

First, we should distinguish between weight (a force) and gravity (an acceleration). Gravity is the acceleration that Earth gives to objects near its surface due to the gravitational force. 

The acceleration of an object near the surface of Earth is due to the combined effects of gravity and the centrifugal acceleration from the rotation of Earth. The resulting acceleration is weakest at the equator and maximum at the poles, with a difference in magnitude of about 0.5%, because the outward centrifugal force produced by rotation is larger at the equator than at the poles. 

The standard gravity, $ɡ_n$ or $ɡ_0$, (the expected or mid-range value of the gravitational acceleration of an object near the surface of Earth) is 9.80665 $m/s^2$. 

Gravimetry measures and studies gravitational anomalies: local variations in topography (e.g., mountains) and geology (rock density) that cause changes in the gravitational field. These gravitational anomalies are measured with instruments called gravimeters. Rocks with lower density, such as sedimentary rocks, result in negative anomalies. 

Another effect affecting gravity is altitude, as an increase in altitude will result in a change in gravity of about 0.03% per kilometer. The maximum difference in gravity is about 2 Gal (0.02 $m/s^2$) from sea level to the top of Mount Everest.

The gravity field of Earth has been measured recently with a variety of satellites (GRACE, CHAMP, GOCE). The resulting data can be used to generate global geoids such as EGM96 (geoid calculator). Gravitational anomalies can be extracted by calculating differences with respect to the geoid. The range of variations is of order ±300 $mGal$ (±0.003 $m/s^2$). 

Other factors have a small effect on gravity, such as the gravitational force from the Moon and the Sun that causes the tides. The fluctuations associated with these effects are of order 0.2 $mGal$ (2 $µm/s^2$).