I use a few different tools and tricks to make Twitter work for me professionally.
The first thing I do is archive everything I tweet using ifttt.com and Evernote. I think it's useful to have a record of what I've said, just for later personal reference and so I can trace any useful things I might have said or shared and not bookmarked at the time. It used to be very straightforward to link Twitter and ifttt.com (ifttt stands for "if this then that" and is a service that lets you link triggers in one place with actions somewhere else), but sadly Twitter withdrew the ability to join the two directly. For the time being you can still use RSS feeds to suck Twitter content into ifttt. The RSS feed for any given user looks like this: https://api.twitter.com/1/statuses/user_timeline.rss?screen_name=USERNAME, where you replace USERNAME with the obvious. Using this as the trigger it's straightforward to save all tweets to an Evernote note, which records their time and date and the tweet URL.
I also use ifttt to save tweets with certain content as bookmarks in Diigo. There's an option to do something only with RSS feed items that contain certain text, and I have triggers set up for certain hashtags such as #speccolls, #archives, #manuscripts and #libraryhistory. Whenever I tweet something with any of those tags the tweet is saved to Diigo with various tags of my choosing. I include the tag "tagme" so that I can find them all on Diigo later and add further specific tags as necessary. This is, unsurprisingly, the bit I am least good at. I need to make some time every week to tidy them up a bit.
I find interesting #speccolls content mainly from two Twitter lists I've set up: spec colls people and institutional spec colls. Every now and again I'll search Twitter users for appropriate new additions to the lists (as well as adding users I stumble across in every day tweeting). I don't follow everyone on these lists, but I keep the lists open in Hootsuite to keep an eye on what's going on, and retweet (with the hashtags mentioned above) what interests me.
Lastly, I use Scoop.it to draw together links connected to special collections. It's set up to catch my tweets with relevant hashtags (as mentioned above), as well as taking posts from a few special collections blogs. Then I choose the stuff that I want to include, add extra description, select a picture to show and add tags. I do this once a week at the weekend, and it takes surprisingly little time. It's a really good way to review what's been happening recently and to get a chance to read in more depth the things I might have skimmed over previously. Hopefully the links are useful for other people, too! I tried using paper.li to do something similar, but it's a completely automated process. I can see how that's useful in some situations, but for me it means I don't come face-to-face with things in the same way, and I didn't find it was picking up all the stuff I wanted it to. That latter could be fine-tuned with a little effort though, I'm sure.