Both books have themes of growing up and death, more specifically, unexpected death at a young age, which I know is obvious, but I might as well say it. Another thing they have in common is smoking. But, the view on smoking in Looking for Alaska is much different from The Fault in Our Stars. In Looking for Alaska, smoking is viewed as "normal" and a way to "fit in". But in The Fault in Our Stars, it was discouraged and simply used as a metaphor both by the characters and the author. For example, Hazel got upset when she thought Augustus smoked. Both books involve at least some discussion of religion; Looking for Alaska when Pudge has to write a paper about religion, and The Fault in Our Stars when Augustus asks Hazel if she believes in an afterlife and when Hazel's dad talks about the universe enjoying being observed. Also the support group meets in a church, so there's that as well.
In Looking for Alaska, Pudge seems to base his life off of Alaska's death (modeled by the chapters: 100 days before, the last day, 7 days after, etc.). Pudge seems to be unable to separate his own life from her death, while in The Fault in Our Stars, Hazel deals with the death of Augustus quite differently. She tells her story building up to his death (focusing on his life), and after she tells of his death and a few important events after, she skips to present day (hence the last sentence being in the present tense) to say that even though Gus is gone, she still loves him. She is able to separate her life from his death in a way Pudge wasn't able to with Alaska, while still indicating that their love survived even his death. Another key difference between the two books is that while Looking for Alaska deals with finding your place in life, The Fault in Our Stars deals with finding your place in death. In Looking for Alaska, Pudge and all of his friends are trying to grow up. In The Fault in Our Stars, Hazel is trying to find a way to cope with the imminence of her own death; and rather than growing up (as she should be), she is having to deal with the process of dying.