Both films have the same basic premise. The protagonist takes something that doesn't belong to him, and then is relentlessly pursued by a demonic antagonist. The Coens deftly use both Leonard Smalls and Anton Chigurh as symbols.
You don't have to be a film expert to realize these two characters represent pure evil. On a deeper level, the Coens use the bounty hunter and the hitman to symbolize the evil of the times, which happens to be the 80s in both films, catching up with America. Or something like that. The Coens do love their symbolism. I'm certainly open to other interpretations.
There are other, smaller similarities as well. Both movies are set in the desert Southwest. Both films have friendly police officers, and finally, both movies feature lengthy scenes with dogs chasing the protagonist. There are more, but you get the point.
Now, having tried to make the case that Raising Arizona and No Country for Old Men are basically the same movie, I want to give the Coens their due. These cats are auteurs, so I'm pretty sure they did it on purpose. Even though the films are quite similar, the tone is completely different. Raising Arizona was made pre 9/11, and the movie is essentially a comedy. NCFOM was made post 9/11, and it is most definitely not a comedy, unless you tend to the very dark. I have a feeling the Coens made the same movie to illustrate how different our world has become.
Of course, this could all be complete nonsense. I can't help myself. I watched a lot of Siskel and Ebert growing up.