What do you like best?
Issues management, Repo projects, Wiki, Repo Discovery tab in the profile page, Github pages and also Gist just to name a few.
The contribution chart and repo analytic graphs is also interesting to look at. The ability to star repos and follow other user led to me discover so many project otherwise goes unnoticed! Also trending projects are very good to know as an open source enthusiast.
What do you dislike?
No three view for each repo. The workaround is an extension like Octotree which will give you exactly that. But it doesn't work well for private repos though.
Make more private repos available to free users, maybe 5? I pay $7 monthly mainly for this feature, but competitors like GitLab offers free private repo for everyone!
I feel like the help pages are a little bit dusty compare to other parts of the site. But I don't really need to go there though, except when I need some special features (like how to sign commits with GPG)
Recommendations to others considering the product
If you're a programmer and not using Github, you're doing it wrong! Github makes it easy to share and view others code so you can learn better. No to mention the whole open source scene relies heavily on Github too!
What business problems are you solving with the product? What benefits have you realized?
Team programming collaboration, issue tracking and project management. It's very fast and easy for everyone to use, since mostly everyone has it. It also makes developers want to write documentations, whether it's an easy README.md or a whole Wiki on the project, since Github display them nicely.