It's tough from a cinematic standpoint.
Alien is not really a sci-fi movie. It's horror in space. pure and simple. And it brought back the genre from the 60's. So it gets a lot of respect for that. I'd say that cinematically it's the better movie and in many ways Carpenter's Thing remake is a rip off. Surrealist painter HR Geiger is fucking amazing and his conceptual designs totally blew away movie goers.
That being said I have to go with the Thing because it is based on the original Thing, which you wouldn't have Alien if it wasn't for. It really wasn't sci-fi either, and it really wasn't a good movie cinematically. But it was one of the best that created the genre of space horrors of the late 50's and 60's.
What made the Thing so successful was because of the underlying fears inherent in that movie and all other such ones of the day. The first is the fear that technology is going to get us into trouble. That scientists are playing with things that they don't understand.
The second is the fear of communism. The Thing was an enemy among us that we couldn't recognize, because he looked just like us- only he was against everything that we hold dear. Those Cold War fears were locked into the psyche of every American and hung over them every day.
Even better was the Body Snatchers, another movie that wasn't so great, but a brilliant underlying idea. It took the same basic idea of the Thing, but it dug deeper. It wasn't about Communism, as many people think. It was about small town lives and small town values in an ever changing and expanding world and the loss of American innocence and the uncertainty of the future. That's the real heart of terror.