What do you like best?
The best about Monodevelop is... that it actually works on Linux and lets you develop ASP.NET applications with C#. Debugging when process is running works fine. I had to downgrade to older version of Monodevelop so I could run server process to host my ASP.NET application and debug it, but overall experience wasn't bad. Monodevelop gives you somewhat similiar experience to Visual Studio with smart advices when you code and showing errors live.
What do you dislike?
Installation of more recent versions was bad experience. Also for my ASP.NET 4 MVC application to work I had to add MySQL provider entry to some global configuration file hidden somewhere on filesystem. There were also problems working with ASP.NET server and I had to switch to older version of Monodevelop so I could run and debug my application.
Recommendations to others considering the product
If you plan to work with latest ASP.NET then you can probably use any editor with addons or Visual Studio Code. I would recommend trying Visual Studio Code first, because it's faster and not so heavy editor. Also, it's supported by Microsoft which stands behind its technology.
What business problems are you solving with the product? What benefits have you realized?
Most of my experience with Monodevelop comes from creating and working with ASP.NET MVC 4 application from scratch. Requirement was that it will be hosted on Linux machine so I also had to make sure it works there. So I've chosen to develop this application wirh Mono and Monodevelop on Ubuntu operating system. Monodevelop was only application I've found working and letting me to do my work, but it involved dealing with lots of problems I've had to solve on my own, because web lacks information, articles, tutorials on how to deal with various problems. Some of my questions regarding Mono and Monodevelop still remain unanswered on stackoverflow.