What do you like best?
It's ready to go!
What do you dislike?
It's really heavy, sometimes bloated with unnecessary options, and not really optimized for the platform was built for. Mobile devices have really limited computing and memory available, and jQuery mobile uses those resources quiet heavily, especially during page transitions and push call.
Laggy animations and jumpy keyframes are almost inevitable.
Recommendations to others considering the product
There are plenty of lighter and powerful alternatives, but jQuery Mobile is packaged with everything you need. I would recommend to use it as a fast prototyping tool, if you don't really care much about performances.
What business problems are you solving with the product? What benefits have you realized?
Building a consistent UI for multiple Mobile platforms The theming system of jQuery Mobile helped me a lot to maintain that consistency and keep the user comfortable in a familiar environment, independently from the device he was using.