What do you like best?
Best thing about MySQL is, It's quite easy to install and use. By using some third party tools you can make it even more cooler. e.g. mk-query-digest & mydumper. Recently, it has gained more popularity as it's being used by loads of major corporations like Twitter, Alibaba, Amazon etc.
What do you dislike?
Most of the people things it's free, but it's quite expensive too. Which depends on for what you are planning to use it for. It could range from FREE to $15K+.
It somehow lacks in scaling, when it's forced to do too many operations at given time.
I don't know why MySQL is trying to do everything, which it should not. It's dependency on Addons sometimes makes task complex and hard to track and maintain.
Recommendations to others considering the product
MySQL is a free-to-use, open-source database. It helps us in effective management of databases by connecting them to the software. It is a stable, reliable and powerful. Quite suited for low & mid-level organization.
What business problems are you solving with the product? What benefits have you realized?
Personally, i have used MySQL in development of several Web & desktop based application. Additionally, i have also experimented lot using it. Being it's open source gives you lot of flexibility. It's community driven development gives us exposure to lots of third party tools which has made my work easier.