Let me explain both concepts...
The traditional method through the web browser still works (although you cannot use the YouTube website to access the videos). To watch YouTube videos you go to Facebook and search "SwitchBru". Tap on them and click on a link on the post that says something like "YouTube on the switch" and it should take you to a site which loads those videos (the site is a pain on the kneck since the switch doesn't display everything correctly but you can at least see the first 3 searches... Barley).
The second method that uses Homebrew to access the site requires quite a bit of work. You require to catch streams from the YouTube site, scan the YouTube site for videos, need to have a decoder, and would need to have a video player on hand built-in. It gets very complex when you get into detail but here are the pros and cons:
Pros:
-Can support any video quality
-Very unlikely that it wouldn't work on any switch version
-Is a lot easier and faster to access
Cons:
-Very hard to make
-May be a large filesize
-Streams may be laggy if you use a 2.4Ghz internet connection (since the switch limits the connection heavily: About 2Mbps on any connection speed)