Update: I figured out a solution. We'll see if it holds up, but hopefully it will work for everyone else as well. I'll just explain what I did in the order that I did it. Since YouTube and other videos in QuickTime and Safari are working perfectly for me now, I have no way of knowing whether all if these steps are essential.
I'll post quick steps at the end of this post, but I'm posting the entire explanation in case it helps anyone to figure out the problem, in case it should return, or help anyone else think of solutions to their problems if this does not work for them.
First, I should note that for some reason I started experiencing the exact same problems over cellular, ironically today (ironic because I've had problems with YouTube over WiFi since September 21, day 1, and video over cellular always worked perfectly until I posted about the WiFi issue-there's a lesson about not bitching and being grateful for what you have, somewhere in there). So, turning off WiFi was no longer a solution. I turned WiFi back on, and still had the same problems, couldn't play video. So, it looked like I was going to be making a trip to the Apple Store, as I was unable to watch video at all.
I decided to keep WiFi on and disable cellular data, and it worked. The same video that I was unable to watch over WiFi or cellular was now playing over WiFi with no pausing or buffering. I tried other videos including videos I had watched and videos I had never watched before to confirm.
Great, so now I had WiFi, but was going to have to disable cellular data every time I wanted to watch a video, and re-enable it for when I need a data connection and am outside of WiFi.
But, to my surprise, I re-enabled cellular data and now video works (through QuickTime/Safari and the YouTube app) both on WiFi and cellular. My guess is that either there was some sort of interference between cellular and WiFi (like maybe something similar to the Verizon WiFi bug in iOS6 where people were mistakenly being charged cellular data while on WiFi, obviously cellular data was also functioning while on WiFi, either in a duplicate capacity, or just sending data usage to Verizon-I am on AT&T), or maybe turning off cellular data cleared some sort of corrupt cache, or something of that nature.
Hope this helps, here are the quick steps (as I said, I have no way of knowing if all of these are essential-likely only steps 6-8 while on WiFi are necessary):
1. Enable WiFi and Cellular Data (I assume that you have both enabled by default).
2. Disable Wifi - Settings>WiFi, choose Off.
3. Try watching YouTube Video.
4. Enable WiFi (same step as 2, choose On).
5. Try watching YouTube Video.
6. Disable Cellular Data - Settings>General>Cellular, choose Cellular Data Off.
7. Try watching YouTube Video. Now, it should work on WiFi (or, at least it did for me).
8. Turn on Cellular Data (same as step 6, choose Cellular Data On.
9. Enjoy porn.