In my edition of Bradbury's Fahrenheit 451, Bradbury himself writes in his "Coda":
There is more than one way to burn a book. And the world is full of people running about with lit matches.
Bradbury paraphrases from the novel Fire-Captain Beatty's explanation of how book burning got started:
"...books were burned first by minorites, each ripping a page or a paragraph from this book, then that, until the day came when the books were empty and the minds shut and the libraries closed forever."
Bradbury gives numerous examples of merely his own books that someone has attempted to censor in an effort to eliminate anything that in any way could be objectionable to anyone.
Bradbury, himself, then, sees censorship as an important similarity in the novel to "our world." And the book was written, apparently, at least in part, to expose the dangers of censorship in our society.
To me, the major similarity (though I do not think it is all that close of a similarity) is that we have become something of a busier, more short attention span society.
Over the years, technology has, you can argue, shortened our attention span. We are used to TV instead of books. Now we are used to Twitter and texts instead of other, longer, forms of communication.
In addition, you can argue that we are also more concerned with material goods than with our spiritual or moral well-being. Like Montag in the subway, we are constantly bombarded with ads. Like Millie and her walls, we want the next kind of technology.
Now, I don't think this has made us bad like the society in the book. But if you have to draw a parallel, that's the one I see.