I understand why an organization would link their Facebook Fan page with their Twitter account to share updates. Time and staff are limited. It’s easier to update once and let the auto-posting magic take over. In my humble opinion, this is not a best practice. I think Facebook and Twitter are two different beasts and should be treated as such. Here’s a few reasons why:
1. It’s obvious you didn’t write for this platform so you look kind of lazy: Posts sent from Twitter to Facebook have usernames (@someone) and hashtags (#something) that are useless and mean nothing to Facebook. Facebook posts are usually too long for Twitter, so they end up with that “fb.me” link back to the Facebook page. If you want visitors to take a certain action (say visit your website), why make them go to Facebook in between? You’ll lose some percentage along the way.
2. You are losing impressions if you post from Twitter to Facebook: I’ve been seeing this in my Wall stream lately: “See XX more posts from Twitter.” That means I have to take an action (click) to view these posts. If the last few Tweets sent to my Facebook News stream before this one are not so engaging, I may not click that link at all. The post you carefully crafted for Twitter may not be viewed by many of your Facebook Fans simply because it came from Twitter.

Twitter posts are collapsed by Facebook
You are also likely losing impressions by not updating manually because of EdgeRank (the algorithm that Facebook uses to determine what to show in News feeds). You’ll get a lower EdgeRank score when using third-party applications to post to Facebook (more on EdgeRank here and here).
3. Facebook to Twitter often leaves out important info: One organization I’m a big fan of recently posted on Facebook to “view the Note below” for info on a new feature on their website. But after it was auto-posted to Twitter, there was no link included back to the Note on Facebook. After they created the Note, it was automatically posted to their Facebook Wall. In a separate status update they instructed to look “below” to read the Note. But only the second update made it to Twitter and there was no “below” to read (except other people’s Tweets). Important note: most people read your Facebook status updates on their News feed and not your Fan page (check your Facebook Insights to see the difference between your Impressions versus Wall visits), so if you refer to something “below” it might not make much sense even on Facebook.
4. You lose the power of tagging: One great way to both get the word out about your page and be a good social media neighbor is to tag other people & organizations in your Twitter and Facebook updates. Since they use different formats, posting from one to the other is not compatible for tagging. On Facebook, all those Twitter “@” tags don’t work and look kind of silly. On Twitter, the tagged name may not display the way you may have preferred and will often use up more of your 140 characters than needed. When auto-posting, you can’t create these important links to other Twitter accounts that share info and help grow your followers.
5. You might accidentally do this:

Why you shouldn't link Facebook updates to Twitter!
Sounds pretty tasty up until the “Flash Fried Baby!” There is a good chance the next word was “Spinach” but we can’t be sure unless we click-through. I’m guessing no one wants to accidentally send out a message like this to their followers and fans. And they certainly don’t want a local newspaper writing about it on their blog: Why Restaurants Shouldn’t Cross-Post From Facebook to Twitter by Robin Wheeler @ RFT
Do you have a different opinion on linking Twitter and Facebook? Are there other good points I didn’t mention? Let me know about it in the comments.