Query being made on behalf of someone else:
What might cause a 1-year-old iPad to suddenly stop being able to play videos attached to email as a URL link — while at the same time still be capable of playing YouTube videos?
An older iPad has no such difficulties.
(I know nothing about the ins and outs of iPads, therefore I ask.)

I would need to know a lot more about the videos and their format before I could even begin to speculate intelligently on an answer to the question. If these are Flash videos the iPad cannot and will not handle Flash of any variety (Apple will not let Flash on iOS devices and at least partially as a result of that, Adobe is dropping support of Flash on all tablet and smartphone devices.)

Another piece of information for you, the latest release of the iOS operating system is compatible with any iPad. Some hardware features may not be available, but other than that your friend's iPad is capable of being upgraded to the latest iOS 7 and the upgrade is free.

I'm going to let the user's local Mac user group try to sort it out.
When 2 iPads in the same household behave differently, it's either something dire or something simple like system settings/preferences (or so I would imagine).

The original iPad cannot be upgraded beyond iOS 5.1.1.

However, a "one year old" iPad should be capable of running iOS 7.

iPad issue resolved by going to settings, then Safari, then delete cookies and new data. All fixed.