Venus appears to be the closest to Earth in mass, density, size, etc. - though they clearly have different atmospheres. Why do Earth and Venus have different atmospheres?
Note: This response is bereft of references. I'll try to add them later.
There is no definitive answer. There are a lot of conjectures, however.
Whatever the cause, the Earth is markedly void of carbon compared to Venus. The amount of CO2 in Venus atmosphere corresponds to a 0.88 km thick layer of carbonate. The Earth's geosphere (lithosphere plus oceans plus atmosphere) collectively contains about half that amount of carbon, almost all of it locked up in the lithosphere. The huge amount of carbon in Venus atmosphere versus the paucity of carbon in the Earth's atmosphere is the primary factor that distinguishes the two atmospheres.
Planet formation. The leading hypothesis regarding the late formation of the Earth is the giant impact hypothesis. Multiple simulations suggest that this collision had to be rather oblique. An oblique collision between a Mars-sized body and a not-quite Earth-sized body would have drastically changed the Earth's rotation rate prior to and after that collision. If the Earth was rotating very fast prior to the collision, that fast rotation combined with the collision could have resulted in the Earth losing a large chunk of its primordial atmosphere.
Venus does not have a large Moon. That suggests that it's formation was a bit less tumultuous than the formation of the Earth. With no big collision to eject that primordial atmosphere, Venus may well have been operating in runaway greenhouse mode from the very onset.
Water. Whether Venus first had free water on its surface or in its atmosphere has long been a subject of debate. Venus is now nearly devoid of water. If water was present in Venus early atmosphere, it would have served to even more strongly magnify the already huge greenhouse effect of a thick atmosphere that is opaque in the thermal infrared frequencies. If the very young Venus did have liquid water on its surface, it wouldn't have lasted very long.
Plate tectonics. One of the consequences of a very thick atmosphere and no liquid water is no plate tectonics. Venus has a very thick greenhouse atmosphere, which makes for a very hot surface. The high temperature of Venus's surface means the surface healed itself too rapidly. Plates couldn't form on Venus (Bercovici 2014). Water is an important lubricant for plate tectonics, particularly for subduction (Mian 1990). (But also see (Fei 2013) for an opposing view.)
With no plate tectonics, there was no mechanism to bury carbon inside of Venus. Plate tectonics developed fairly early on in Earth's history. By the time the Sun became hot enough (the early Sun was faint), the Earth had already started the process of sequestering away carbon into the lithosphere.
Life. Life loves carbon. It is one of the key agents by which atmospheric carbon is transferred to the lithosphere. Life apparently never had a chance on Venus.