What Content Should You Post on Website, Twitter, Facebook?
Your website is your real location--think of it as your online office. This is where the "official" version of everything belongs.
Your Facebook page is a place to socialize--think of it as the lunch room, coffee shop, or after-work watering hole. This is where people will engage in "light" conversation and banter. It is a good place to post links and news of interest in your field, but which are not directly related to your organization. Important conversations (such as meetings, policy discussions, and so on) probably do not belong on a social website, and are better handled on a forum or blog discussion on your website. Although you can announce that those discussions are happening via Facebook, with a link back to the actual discussion.
Your Twitter feed is a place for timely announcements--think of it as an announcement over a PA system.
Let's take a typical example of something that could involve all three online services, and see how each could play a role that compliments the others without pointless duplication. Say you are hosting an important event - an annual conference:
Website: your website should host the official event calendar and description, possibly with online registration and other event-related services.
Twitter: this is for timely announcements, so use Twitter to send out short notices for things like registration openings and deadlines, changes in itineraries or agendas, and perhaps a welcome and thank you note at the actual start and end of the conference.
Facebook: add a brief event announcement, which points back to your official web page. You can make this a Facebook event if you like, but that is optional since you aren't really using Facebook's calendar services. Pre-event announcements are helpful to build interest, but Facebook really excels at post-event socialization, such as posting results, awards, photos and albums, or videos. These are the sort of things that people enjoy reminiscing and bantering about in a social medium like Facebook.