What do you like best?
The reports are highly customizable and easy to work with. It also filters out referral spam, which is something Google doesn't do.
What do you dislike?
The interface feels a little rough around the edges. I'd also like to see it use a front controller design pattern that keeps code besides index.php out of the webroot, but these are things that can be improved down the road. I'd also like to see a more attractive default theme. As is, it's a little on the ugly side.
Recommendations to others considering the product
It's very easy to install, so just give it a try and see for yourself.
What business problems are you solving with the product? What benefits have you realized?
I'm trying to track and maximize conversions. Piwik provides detailed information on individual users, which Google only allows in aggregate. This allows much more accurate generation of narratives to describe the desires of users on the site.