I am able to play YouTube videos on all of the test systems (some better than others, but they all at least start playing), and I haven't received any similar reports. The systems here are all 10.4.11, and my Internet connection is much slower than yours, so I suspect this is a local problem.
Please make sure you've disabled any adblock or privacy add-ons first. Some of these will interfere.
If temporarily disabling them doesn't fix your problem, next try a clean profile to see if a stuck or improperly configured setting is at fault. See "How to reset your profile" at the lower right.
If neither of these work, we can look at some other possibilities which are much less common.
I use TenFourFox FPR5 with Tracking Protection (from Mozilla) on a G4 1.33 GHz with 1.25 GB RAM, and VP8 videos at 360p work well. I will usually wait for most of the video to load in cache before playing it. Also, using either the classic YouTube (known as "disable polymer") or loading just the embedded player (not the rest of the page) gives me the best results. I hope this help!