For example, when Taylor Swift plays her song "My Tears Ricochet" there's essentially two things that separately copyrighted: 1) The musical notes and words in that particular order, and 2) the performance itself (e.g. Taylor Swift performing it). [1]
Using the performance itself is what YouTube typically has a licensing arrangement for. If you weren't on YouTube, you'd need to license it separately through ASCAP and pay them directly. Also, when you do license it you don't have carte blanche rights to do whatever you want with it, the license will come with limitations on where you can use it ("sync rights"[1]), both technology-wise (e.g. TV, radio, internet, mobile, etc) and sometimes even geographically (e.g. North America, Europe, USA/Canada, etc).
In order to perform the work yourself in public, you actually need different permissions called performing rights [2]. These licenses will typically allow you to perform the work using a musical instrument and/or your voice. You'll often pay different amounts if you charge for your performance or not, if the performance is professional versus amateur. Also, you're not typically allowed to reproduce the sheet music or lyrics physically, either for sale or to give away. I suspect you'd also pay more for imprinting the words on a screen (e.g. sing-a-long style).
All this complicated licensing is not part of the copyright law itself, it's separate is really more about contract law. Copyright law allows the copyright owner to assign rights, the owners have elected to assign them in a way designed to extract the most amount of profit from their asset.
I'm not saying I like the way the copyright system works (I don't), but that's how the law is written.
Also, I'm not a lawyer...just someone passionately upset about copyright and patent law.
[1] https://en.wikipedia.org/wiki/Synchronization_rights [2] https://en.wikipedia.org/wiki/Performing_rights