I finished Season 1. I've been reading the old thread too, following the links and taking in the responses. I'm having a hard time processing it. Ironically some of the comments foreshadow the backlash to Girls (e.g. ronito and Brian Fellows sound like some of the criticism of Girls). I almost don't understand how anyone can like Enlightened and not appreciate Girls (except in the hipster sense that Girls came afterwards and took all the shine). At least Girls provides some laughs... I'll save the rest of my thoughts until after I completed Season 2 ― unless anyone wants to go over the Girls/Enlightened comparisons (shame anaron is banned :-( )― but a small shoutout for the Consider Helen episode. I had heard a bit about that episode be before watching and it definitely delivered on the hype.
I will say that Enlightened steps up its game significantly in S2, but I think Season 1 is pretty great as well. I'll add that I think that Enlightened asks far more significant questions (even if it does in an anxious-bordering-on-mentally-ill way, and is never able to fully answer them). I've never seen a show do "malaise" or discomfort the way Enlightened does. I think it has a very strong social conscience, in a way that few shows do. It's not quite righteous and targeted the way Sorkin writes, it's fuzzy and unfocused and confused. It's lonely. It's lost. Really really interesting stuff. It's so calm but so scared. I think it does a remarkably good job of portraying the thought process of a person who sees themselves as an "agent of change". Actually, it's had odd parallels to my personal experience, as I'm finishing up a master's course on protests against globalization, and the way the authors I read speak about imagination, and the really fuzzy character of their proposals, their lack of concrete ideas, just the sense of discontent with the world and the need to make it better... White seems to have a very very good grasp of what these people are like.
I do enjoy Girls, but I think the first season was stronger than the second, and the second is a little too self-aware.
Both share strong authorial vision, but very different authors. Both have fairly snappy dialogue. They have different tones, though. Enlightened is probably closest to Parenthood in the way it shows conflict between characters. Enlightened is far less cartoonish than Girls, which has a feverish quality. Enlightened feels a little bit like a slow moving train, like it is being hauled inevitably to its destination, while Girls doesn't really have much intra-episode momentum.
That being said, Girls' best episode is Hannah leaving NYC for an episode, and Enlightened's best episode is Levi... err... well, you'll get to it, but point is that Levi is somewhere that Amy isn't. There's something peculiar about delocating a character that gives some essential insight into their personality and their reactions to things. The more episodes I see of more TV shows that do this, the clearer to me that this is a really great technique. House of Cards' best episode was Frank at his alma mater. Louie's best episode this last season was "Miami"...
Edit: Filled out my answer a lot more.