What do you like best?
Asterisk is a great project, full featured and enterprise ready. We use it for all of our conference calling, interoffice communication, and making and receiving outside calls. It's rock solid, Debian based, and ready for deployment. We have put them in lots of places, and never have any issues. You can buy support for cheap and the help they give you is top notch.
It's easily customizable, and works with EVERY device on the market. It's also been around forever and has a solid future.
What do you dislike?
The interface isn't the best, the web stuff is kind of dated. The upgrade/update scheme can be a bit funky if you're not paying attention to newer releases.
A lot of work to get setup, but once you get everything working good it's an amazing solution. A lot of testing needs to be done to see if it can meet your organization's needs.
What business problems are you solving with the product? What benefits have you realized?
We use it to take phone calls, make calls, and host large conference calls in our office. The benefits are great, because it is a real good open source project that you can trust with your infrastructure.