What do you like best?
Mercurial is an Open Source very powerful DVCS. It is at the level of Git but with a much simpler user interface and one which provides a smooth transition from a centralized system like Subversion to a DVCS. It is written in Python and is thus portable: Linux but also MacOS, Windows are fully supported.
What I most like in Mercurial is its very gradual learning curve: even if you are not a version control specialist you can easily set it up for personal or group usage.
However, you can also add powerful extensions: rebase, transplant, bisect, large files, etc. You can use the phase mechanism to decide whether it is safe or not to allow history rewriting, you can install and use TortoiseHg, a very intuitive portable GUI, etc.
Also, a service such as bitbucket provides a web service to collaborate with your friends or colleagues on code with pull requests, as they have been popularized by Github.
What do you dislike?
What I dislike most in Mercurial is that.. it is not more widely use!
Git is becoming very popular, even in companies previously using Subversion, Perforce, or Teamwork. People know git commands and so have difficulties with mercurial differences in: branches philosophy (Mercurial branches are more long-lived, while lightweight branches like in Git are similar to bookmarks in Mercurial), fetch/pull commands which are opposed in Mercurial and Git, etc. If you are heavily using both, like I do, you need to think twice before pulling or fetching! And even more when undoing mistakes, or you run the risk of loosing data.
Besides this lack of popularity which sometimes makes it difficult to work with others, Mercurial does not really have shortcomings in my honest opinion.. It "just works"!
Recommendations to others considering the product
I very recommend the use of Mercurial, one of the most powerful and easy to use DVCS currently available.
What business problems are you solving with the product? What benefits have you realized?
Version Control is a service of paramount importance for any software producing work: being able to merge and revert changes, work together, review changes, find bugs with bisect, etc. With web services such as Bitbucket you can in addition collaborate with other developers worldwide.