The near side and far side have very different topography. Compared to the near side, the far side is far more rugged and has far fewer maria. The near side in places has very thin crust (the maria).
Other than these surface features, the biggest indicator of a disparity in crustal thickness is the two kilometer offset between the Moon's center of figure and it's center of mass. This is typically attributed to a thicker crust on the far side of the Moon, consistent with the observations of the different surfaces of the near side and far side.
NASA discovered that the center of figure (geometrical center) and the center of mass are offset from one another in the late 1960s. The simple act of observing how a probe orbits an object gives clues regarding the interior of that object. Scientists know, for example, that Mars has a partially molten core from observing the orbits of the many probes that have been sent to Mars.
In the case of the Moon, NASA sent 16 unmanned probes to the Moon in the 1960s prior to the Apollo 8 mission. The first four of these, Ranger 6, 7, 8, and 9, were impactors. Their impacts the Moon indicated the Moon's center of mass was displaced from its center of figure by 2-3 kilometers. Later missions confirmed this, and also confirmed that the Moon's near side has five large mass concentrations ("mascons"). The Apollo 15, 16, and 17 missions carried laser altimeters that gave very precise measurements of the distance to the Moon's surface. These further improved the estimate of the Moon's center of figure / center of mass offset.
(Aside: Read Bizarre Lunar Orbits for a description of how those lunar mascons change how things orbit the Moon.)