If can see in safe mode then it is possibly 1 or more Extensions. Should disable and turn back on at a time and test while doing so, also refresh the Youtube page your on.

Lets do this and will know if it is your profile or not.

This is the best way to uninstall Firefox. Then Delete the Mozilla Firefox Folders in C:\Program Files and C:\Program Files(x86) Then restart system. Then run Windows Disk Cleanup. Then run it again and click the button that says Cleanup System Files. Note: your Firefox Profile is saved. But you should make a back up before you do : https://support.mozilla.org/en-US/kb/back-and-restore-information-firefox-profiles

Reinstall with Current Release Firefox 57.0.1 with a Full Version Installer https://www.mozilla.org/firefox/all/

See if any updated Drivers for your Video Card : ATI Radeon HD 4300/4500 Series as the drivers are almost 2yrs out of date. Should download and keep if there is in case need to put them back. https://support.amd.com/en-us/download fyi: Microsoft will and can repalace your Hardware Drivers with Microsoft Generic Drivers. for Video is 2006.

Please let us know if this solved your issue or if need further assistance.

I followed all your instructions, but still could not see videos. My video driver was current. Out of desperation, I reset Firefox, and omg! everything now seems to be working, and I did not lose any of my add-ons, bookmarks, etc. Guess I should have tried that first. No idea what was wrong, but I appreciate your help and suggestions. Thank you.

There is a similar thread here: https://support.mozilla.org/de/questions/1188140 and this happens also on my 64Bit Linux installation of firefox 57.0 and 57.0.1 . I already dumped the complete .mozilla/firefox directory and created a new profile this way: no improvement. Which excludes any addons as culprit. Is still cannot play https://www.youtube.com/watch?v=islcMmfHHqE .