I think Chris Nolan is a better director and storyteller, by far.
Abrams makes good films, but I've yet to see him make a film that I would call truly "great." And in several instances, frankly I feel that he hyped films too much to the point that they were disappointments in terms of the supposed secrets and reveals they were supposed to include. He has the "mystery box" approach, yet too often the box is empty or the contents are precisely what everybody knew they would be far in advance.
I think Abrams is great at directing action and effects and has a nice approach to humor in the midst of action storytelling, but I feel his dramatic approach is good without being particularly impressive. There hasn't been much weight or deeper meaning to anything he's made so far, aside from perhaps (to some extent) Mission: Impossible III (which is also probably his best film, followed closely by 2009's Star Trek).
Here's a simple question: what was Kirk's arc in Star Trek Into Darkness, really? What was Spock's arc, beyond "it's okay to feel emotions?" The villain is the only person with a significant arc and motivation in that film at all, while the main characters and heroes mostly lack much narrative arc at all. I recognize the film plays briefly with the question of whether extrajudicial killings and militarization are good or bad, but it frankly never even sets this up as a serious moral dilemma -- we know what to think from the get-go, and the change of mind comes pretty quickly and with almost no consideration from Kirk, so it's over as a theme within a few minutes.
Of course, pure entertainment is fine, but if that's all there ever is and if it becomes predictable in the presentation and can't live up to its own constant attempts to hype up some big secret or twist that doesn't pay off, then it is going to affect the overall assessment of the filmmaker in the long run.
I get a feeling that he gets caught up more in trying to figure out ways to fool the audience or to do something that will feel clever, than approaching the storytelling with an eye first and foremost toward telling the story that needs to be told. The most glaring example of this approach going terribly wrong is his most recent film, Star Trek Into Darkness. There are so many "See what I did there? Get it?" moments that it gets distracting and gets in the way of the film's progression.
By contrast, I feel Christopher Nolan approaches his films with a much more serious eye toward the storytelling and never wants to let secondary considerations distract from telling the story that wants to be told. I also feel that his talent goes beyond mere entertainment value, and into something far more artistic and thoughtful, and even in his big budget summer blockbusters there are deeper themes to consider if you wish to look that far into it. Contrast the arcs for any characters in a Nolan film to the arcs for characters in Abram's films, and I don't think there's much contest. Again, Mission: Impossible III (a great film) is an exception to this, but most of the time the character arcs in Abram's films don't tend to be extremely complex or hard to predict, and sometimes there just doesn't even seem to be an arc to begin with.
Stylistically, I prefer Nolan's films by a wide margin, but I can appreciate why some folks might prefer Abrams. Abrams' ability to deliver excellent action and to provide character interactions within the context of an action scene are pretty impressive. Some people mock the lens flares in his work, but I don't mind it and overall his films do usually look great. For me, though, Nolan's work has a grace and thoughtfulness, a weight to the settings that is intense yet never overshadows the weight of the performers and narrative. He tells real stories about real people, and he can do so and still make it exciting and entertaining.
As storytellers, Abrams and Nolan approach filmmaking quite differently, and I feel Nolan's approach is superior and delivers superior end results.
*UPDATE: I feel like my answer reads a little too harsh about Abrams, whose films and TV shows I generally like (some I like a lot), and whom I do think is a very good filmmaker. This is a critical analysis intended to note the comparison between the filmmakers, and why I think Nolan is better, so that's why it focuses a lot on highlighting the areas where I think Abram's films have flaws etc. Just wanted to clarify that!More questions on Movies: