What do you like best?
Fast to implement, fast running and easy to learn.
You can setup CodeIgniter (version 3.x) in a matter of a few minutes. It is the most rigid way of doing MVC with PHP. But the great thing is that you don't have to stick to that pattern if you want it otherwise.
I like very much the possibility of creating your own libraries, which is also an easy way of linking to third-party functionality like Mustache, Kint, PHPExcel, TCPDF, and others.
CodeIgniter is great for building REST API's. You simply skip the view in the MVC and you're good to go.
The documentation is sufficient and has tutorials to get you started quickly.
The performance is great and outperforming most other frameworks. Mainly that is because CodeIgniter is not adding a lot of overhead and sticks close to what PHP was supposed to do. An important remark is that PHP 7 is supported from version 3.14 and up.
What do you dislike?
The latest version at this moment of writing is version 3.15. That doesn't support namespaces, which is a big drawback in relation with other newer frameworks like for example Laravel, Slim or Zend. That makes that the reputation of CodeIgniter is currently an old framework that is not matching up with the new ones.
Currently, progress is made with version 4 of this framework, which is going to be a complete rewrite of the framework and which supports many features that are also available in Laravel. But progress is slow and is already taking almost 2 years and is just in an alpha phase.
CodeIgniter does not support middleware (very popular in Laravel and Slim).
Regarding version 3 at its current state, there is not much to dislike if you know what CodeIgniter is not able to do (namespaces, middleware).
Recommendations to others considering the product
If you consider CodeIgniter have a look at version 4 (alpha phase) and consider that it is not backward compatible with version 3 (current version). If you want a fast performing alternative you could also consider the Slim framework, which looks a bit like Laravel, but smaller. If you want to join the majority of PHP developers looking for a framework, consider Laravel.
What business problems are you solving with the product? What benefits have you realized?
During the years we have built quite a lot of Ext JS based applications that were communicating with a server backend based on CodeIgniter. Some of these applications are almost 10 years in operation.
Still when I need to have server backend running fast for a new application I tend to grasp CodeIgniter from the shelf rather than Laravel, simply because I can start programming in a matter of minutes without struggling with the setup of the framework.