It's one of those (rare?) horror films that actually sticks to the dictum of 'less is more' and creeps you out by implication, rather than explicit displays of gore, most of the time. Which makes the gore all the more shocking when it's really used.
In terms of comparison to The Thing, it's hard to say which is more effective as a horror film, I think, because they're effective in different ways. In Alien you don't see much, whereas in The Thing you see *everything* - it's total gore-porn horror maximalism - whenever the organism is actually transforming or attacking someone; but when it's quiescent, you see literally nothing, and the fear derives mainly from paranoia and the uncertainly of who has already been got. The horror is also very much sexual in Alien, all about rape and involuntary pregnancy, whereas it isn't in The Thing, as version already noted. Thirdly there's the location: in Alien there's the a sense of fear for the characters due to their precarious and isolated situation, whereas in the The Thing we're very much on Earth, and it's just good fortune the creature is, for now, confined to Antarctica - but for how long?
Here's a thought, though: both films are ultimately about parasitism, about an organism that invades others, bends their bodies to its will and ultimately destroys them - is it a coincidence both were made around the time that the Aids pandemic was exploding into the public consciousness?