What do you like best?
Malleability is Nagios' true strongpoint. I have seen it used in 3 fairly different environments. I've used 3 different frontends, currently using 2, to interact with the nagios backbone. Each having their own strengths and weaknesses respectively. When it comes to interacting directly with it via the CLI, while not the nicest to look at or work with, it does exactly what you tell it to.
What do you dislike?
Directly editing configurations can sometimes be a bit of a process due to having to dig through the files in the terminal. You get out what you put in, and for most people the quality of their nagios experience will have to do with what front-end they end up putting in.
Recommendations to others considering the product
I have only ever used nagios to monitor systems, would recommend for large scale operations. It is able to handle all of our needs and is highly customizable when it comes to its alerting. I would recommend looking into the various front-ends that are available for it, as that will determine a large chunk of your experience.
What business problems are you solving with the product? What benefits have you realized?
In all the scenarios where I have used nagios, it has been the monitoring of a vast amount of servers/systems and being able to report issues quickly. Without nagios, the online business unit operations teams would be operating "blind".