On Bitbucket you pay for developer and have infinite private projects.
On GitHub is you pay for private projects and have infinite developers.
If one company has many projects and few developers, Bitbucket is cheaper.
I also have small customers who don't even know what a repository is. I self host with gitolite (I don't need a web interface, only push and pull) and give access to the most illuminated of them to a self hosted Redmine server for managing tickets. I write documentation in Redmine's wikis.
That is the legacy pricing model. Their new pricing model matches Bitbucket: https://github.com/pricing
Still, with Bitbucket we can have unlimited private repositories for free up to 5 users, then it's $2 per user. With GitHub unlimited private repository come at $7 per month, then it's $5 per month for the first 5 users and $9 for the others.
I use the public GitHub many times a month. More importantly, every single job I've had for nearly the past decade has stored its code in GitHub. Not publicly accessible, and rarely at a private github.com account; every place I work uses a privately hosted GitHub Enterprise server. I wouldn't even consider using anything else, even though GitLab (and to a lesser extent, BitBucket) offer exactly the same functionality.
GitHub was groundbreaking when it arrived. Some people are having an emotional reaction and defecting to GitLab this month (for reasons that escape me), but GitHub has all the inertia. More importantly, GitHub has a vast community. It's just where code lives now.
I'm curious as to what sort of development you do or where you work that you're not a regular GitHub user. It's inescapable in every software company I've worked at since its inception, and as a consultant, that's a huge number of companies in nearly every industry. The only place I've worked in the past decade that didn't use GitHub was an all-Windows shop that used some horrible old proprietary revision control system whose name escapes me. Do you use git but not use GitHub (public or private or enterprise)?