Paper currency is backed by a government promise. Of course, if people lose faith in a state or a government persistently debases the fiat currency, people will lose confidence in it. But as I've pointed out before, while there have been many hyperinflationary episodes, it's not that frequent of a problem relative to the number of countries and currencies in existence.
Gold has value. It can be used by dentists. It can be used in electronics. It has been inherently valuable for millennia.
As I said, paper currency is too complex a topic to go into fully here. Unlike gold, it obviously is not inherently valuable. You can look at Weimar German marks, Confederate dollars etc. to see that. Or even a US 1978 dollar versus a US 2014 dollar, to some extent. People who think Bitcoins can be worth something because the dollar seems to have some value are going to learn a hard lesson.
This is a moot point. The value of gold is almost totally unrelated to its utility.
"they are worth something because people think they are worth something".
gold and the paper currency is printed on is only worth something because people think they are worth something.