Downloading a file from the internet is not inherently illegal. After all, you need to download the video in order to watch it. (It could be illegal for the server to send it to you, but that's a different issue.)

What could be considered illegal is storing it on disk. This could be considered making a copy, one of the exclusive Copyright rights.

Furthermore, just accessing the site with your tool could be considered a breach of contract, the contract being the site's terms of service. That's not illegal, but a company might be able to seek legal remedies in civil proceedings.

Finally, since YouTube isn't a government actor, it's free to block your access to the site for pretty much whatever reasons it deems relevant. (A notable exception is blocking based on a protected class such as race.)