Although social media can facilitate conversations and relationships, it can also be used pretty effectively as a broadcast mechanism, Twitter included.
One of the truisms of association management is that our members don't pay attention, don't read what we send them, and often don't know what's going on, even when it's something that *should* be really important to them.
Twitter can provide an additional platform to get the word out.
I'm going to call myself out here. I never seem to know when the ASAE calls for presentations for various conferences open and close. Yes, ASAE sends emails and posts the information to the web site, and yet, somehow, I always seem to miss it (see above re: not paying attention, not reading what they send me, and not knowing what's going on, even when it's something that's important to me). You know when I spot the announcements? When asaecenter tweets them out (i.e., Nov 9th tweet about AM2010 call, which I actually spotted and am currently working on a submission as a result).
Guy Kawasaki is the master of Twitter broadcasting. His use of Twitter is somewhat controversial, but he's up front about the fact that he uses it as a medium to broadcast his answer to the question: "What's interesting?" It's not a conversational mechanism for Guy.
This is not to say that it has to be all one or the other - I would hope by now that it would be clear that Twitter can fulfill a number of functions at once - but could your organization use an additional platform to get the word out about what you're up to? Could Twitter be useful for that?