The short answer to what is so moving about this film: go and see it. The longer answer is: great performances, a great story, the desolate beauty of the lunar setting, go and see it. This review, written about half an hour after leaving the cinema, involves a bit of reverse engineering - starting with the conclusion, I am working backwards to understand exactly what left me with such a strong impression.
The film is essentially a two hander led by Sam Rockwell as Sam Bell, the sole occupant of a lunar mining base, and Kevin Spacey as the voice of his assistant robot GERTY. Sam is coming to the end of a three year contract and preparing for his return to Earth, and his family. However the stress has been getting to him, and with only a few weeks to go, a rover accident leads to a series of events and discoveries that completely undermine Sam's understanding of his world.
Moon opens with a back-story - the solution to the Earth's energy crisis rests on mining lunar helium-3 for fusion - that is simple enough to be explained in a few seconds, and plausible enough to support the later twists. The attention to detail in the moon base design, with it's padded coridoors, ceiling tracks for GERTY, rover vehicles and giant harvesters, also helps with plausibility while being a fascinating study in itself. The external lunar landscapes are mesmerizing and the sense that you are transported to a real place is very strong. There are a lot of secrets on Sarong base. Moon never forgets it is a science fiction movie, or that it is a story about a human being, even though at times it takes you into situations as bizarre as those of 2001 or perhaps Solaris. The story and back-story is misleadingly simple given the number of themes it explores on both personal and humanity-wide scales.
The comparison with 2001 and Alien comes from the similarities in look and feel of Moon, particularly the base interior, as well as many of the sci-fi themes. However initial references and similarities often mask very different concepts. Like HAL, robot GERTY is an exploration of the uneasy relationship between man and machine; however GERTY is also an original science fiction creation with it's emoticon screen, and ultimately plays a very different role to HAL. Like the Weyland-Yutani corporation in Alien, Lunar Industries explores the theme of corporate power; but the tale of out-of-sight human exploitation is told much more effectively - it's not just about power or paranoia for it's own sake but about the terrible means humans will go to, and justify, in order to achieve something great.