What do you like best?
Just like any Atlassian products, it integrates tightly with the rest of the stack, giving you a complete development experience.
The ability to visualize what is going on in your code repository is also nice, as sometimes, you just need to be able to look at a piece of code without having to check out the entire code base. You can also perform diffs online in the browser.
It also indexes your code base, so searching through your code (if your code base is large) can be very handy as well.
What do you dislike?
For some reason, FishEye (at least when I was using it) feels very inconsistent with the other Atlassian products. For example, the user management part is clunky compared to other products. It now supports DVCS solutions such as git, but Bitbucket feels like a much better tool to use instead. Maybe it is the product's history, feels more like a tool for traditional VCS such as SVN.
Recommendations to others considering the product
If you are using SVN or CVS, it would get the job done, but if you are using DVCS, such as git or mercurial, you should really consider using Bitbucket instead.
What business problems are you solving with the product? What benefits have you realized?
Mostly as an online code browser to view code without having to check out the entire code base. Diffing between revisions to find changes during team meetings.