Push your Facebook updates to Twitter if you must, but please don’t push your Twitter updates to Facebook. Tweets necessarily employ abbreviations, shorthand, lack of punctuation, etc. due to their 140-character limit. Good tweets also employ hashtags. Sometimes they reference others’ Twitter handles. None of these things translate well to Facebook. Plus, people typically tweet much more frequently than they post to Facebook and also typically retweet the same information multiple times in one week or even one day, which is extremely annoying on Facebook.
Bonus tip: If you do link your Twitter account to Facebook, remember that you did so, and keep your Facebook status updates short enough to make sense as Tweets. Long paragraphs do not work well on Twitter. Sometimes a long update from Facebook will be cut off with ellipses, other times only a link will be tweeted without any explanation text. And if the update is coming from a Facebook profile rather than a public page, a reader on Twitter may have to log in to Facebook to even view what you’ve written. He probably won’t bother.