What do you like best?
At first that framework very compact, flexible and easy to use, if i did't need all features of Laravel i can use Lumen - micro framework based on laravel.
When i use IOC i can change the most default services of framework!
Ususally i choose my own auth module and all what i need to do - just load my module as is and change 1 string in configuration file, it's awesome!
All modules in laravel is stable, flexible and easy to use and rewrite.
Laravel have a huge community in all countries. I'm from Almty/Khazakstan and in my small town we have laravel community! We share own libraries, experience and projects.
What do you dislike?
I have some troubles with elexir - i think need more maluals for it.
Recommendations to others considering the product
At first read the docs and subscribe to laracasts.
Learn MVC and IOC design patterns that can help you at start.
Create git repository and easy progect for experiments!
If you want more - try explore Illuminate source code! It contains a lot of contracts and best practices wich you can use in your own app.
Explore hexagonal (Oninon) architecture - use domain logic and application layers - thats make your code stable and flexible!
And at end learn open-source laravel projects like October CMS
What business problems are you solving with the product? What benefits have you realized?
My first laravel project was user mail subscribe system, it based on laravel4. That system work whithout troubles about two years!
My second project - web traffic arbitration system it in develope now.
And third project - backed system for self-hosted CRM based on OracleSql with Delphi desctop clients and web clients. It based on Lumen.