Just so you get a better picture of what happens when your phone is accessing Youtube, it's essentially just a matter of your phone is connecting to some online server. Looking at from your point of view, it's a basic network -- two devices transferring data between each other over a network. From a more detailed point of view, what's actually going on is one of Google's online Youtube servers is sending data down a very twisted series of servers (in different locations and different countries) and cables (some buried in the ground, some lying on the bottom of an ocean) that end up going to one of the online servers run by your ISP (Internet Service Provider), and eventually arriving to your smartphone. People often gripe and whine about lags and problems with their Internet connection but they fail to realize it's really a miracle it works as well as it does. But getting to the matter of using a VPN, in that case you're bypassing the direct connection your ISP has in place, and instead you're indirectly and intentionally using an alternate networking path to those same Youtube servers. With a VPN you'll often hear the term 'tunneling' as that is what you're doing, the 'tunnel' is that alternate network path.
When you're using your VPN service, your smartphone is still connecting to Google's Youtube servers, but simply using a different networking path to do so. The issue you're probably experiencing is some issue with your ISP -- you can access Youtube via your VPN, you cannot access Youtube via just your ISP. Since your brother's WiFi isn't having this problem, is he using the same ISP as you?