Avid NASCAR fans at the Sharpie 500
Following a company on Twitter is unique. It gives a interactive approach to marketing. The company (if doing it right) will interact with you, the consumer/fan. If you tweet to the company, they will respond. This gives you the consumer/fan warm fuzzies as you have just elicited a response from the big company. It implies they care about you. It goes even further, you see if they company responds it shows they are there to actually interact rather then just push out marketing. When they are just pushing out marketing they are turning Twitter into somewhat of a traditional marketing platform. This defeats the purpose of using Twitter in a way.
Becoming a fan on Facebook is somewhat less one-on-one than Twitter. It still is interactive and still gives you the ability to reach the company. But when a company posts something, you plus many other fans will probably like an item or write a comment. Most companies are not going to respond to each individual comment; where on Twitter it makes more sense to do so. On Facebook, media such as video, pictures and other forms of media can be viewed directly from your live feed. You can visit the fan page and see how many other people like it. You can see how other fans are actually interacting with the company. It is in a way, somewhat more robust then Twitter.
Now that I have given a slight compare-and-contrast session, lets answer the original question. When someone asks me if they should both become a fan on Facebook and follow on Twitter, I say absolutely. I actually have multiple reasons for saying yes. On Facebook the post from the company will stay in your news feed for nearly 3 hrs (if you scroll down far enough), while on Twitter it is instantaneous. If you are not on Twitter at the time of the tweet, you are more then likely going to miss what was said by the company. That is unless the tweet was directed at you in a @reply. That is the largest reason.
The other reasons are smaller but still important. On Facebook you get a longer-lasting interaction when commenting on a post/photo/video on the companies fan page. On Twitter it is a one time-shot and unless they respond it is over. With Facebook, you end up interacting with other fans! That is one of the best parts.
So if you use both Twitter and Facebook, consider becoming both a fan and a follower of your favorite companies, blogs, or celebrities.