I first checked the video in IE. After installing the flash plug in for IE, I found that the audio and video played fine. So, it wasn't a YouTube issue at all. Checking again in FF3, it still didn't play. Time to search the forums at Mozillazine.
I eventually came across this link after doing a search. It turns out that the issue was the way Vista, my OS, interacted with Firefox 3. I'm copying the steps here for Vista and Firefox users just in case someone comes across this issue and Googles it.
Windows Vista volume mixer
- Open Firefox and visit a site that should play sound.
- Click the volume icon in the Windows task bar.
- Click Mixer. The Volume Mixer window will appear.
- Make sure the Firefox slider isn't at the bottom.
Now playing on VLC: Nomo - Brainwave