Why is the book An Imperial Affliction so important in Hazel's life?

Because that story is similar to her but has no clean end and ends in the middle of a sentence. So Hazel is curious to know what happens, what happened to Anna, did she die, what happens to her mother etc. Which was clear in the movie itself:

                     HAZEL
       What we need is a sequel.

                     GUS
       Yes. We need to know what happens
       to Anna's family after she dies.

                     HAZEL
       That's what I kept asking Van
       Houten for in my letters.

                     GUS
       But he never wrote back. 

IMSDB

It's a dying girl's wish to get a closure on her fav story which is parallel to her but ends abruptly. Getting those answer might help her tackling her situation too.

But remember neither that book nor the writer is real:

An Imperial Affliction is not a real book, and Peter Van Houten is not a real person. However, An Imperial Affliction is in some ways based on two books I love. The first is David Foster Wallace’s Infinite Jest. Most of the references Hazel and Augustus make to AIA are related in some way to something from Infinite Jest, and I wanted readers of IJ to be able to make those comparisons. But Infinite Jest is not about cancer. Peter De Vries’ amazing and beautiful and hilarious novel The Blood of the Lamb IS about cancer, and most of the broad observations that Hazel makes about An Imperial Affliction — how it is a book about cancer without it being a cancer book, how is is funny and respectful and reflects the reality of experience in a way she has rarely encountered—come from my own experience reading The Blood of the Lamb. I can’t make An Imperial Affliction real. It’s not the kind book I could write well, and on some level, the thing that we imagine will always be better than any real approximation of it that might come to exist. But if you wish to read An Imperial Affliction, I’d encourage you to read Infinite Jest and The Blood of the Lamb and then try to blend the feeling of those two books. — Bustle