If people decided tomorrow to suddenly stop accepting Dollars. Then a Gold standard dollar would be able to be exchanged with the US government for Gold while a Fiat dollar would only be usable to pay taxes.
If a fiat dollar is accepted as payment for taxes, then it will be accepted by those who have to pay taxes, and by extension, those who deal with former group, and so on. In fact, that (tax liability) is the foundation of the acceptance of money in domestic exchanges. So it would be unlikely that people would stop accepting fiat dollars while the government still has power.
This is the pro side of the Gold standard. The con side is the value of Gold standard money is tied to the value of Gold.
What would be the point then? We can already buy gold with dollars, though the price of gold fluctuates. If you look at the price of gold, it is much more volatile than the CPI. The reasoning of goldbugs is that gold is a stable store of value, but it clearly is not stable at all.
As long as the economy values the IOU it will be traded back and forth in exchange for services and the only pros/cons between the standards is what the government would be doing with the flexible value of Fiat money.
What the government can do is quite a lot - smoothing the business cycle if it is competent. If it is incompetent, then it can make the business cycle worse, prolonging recessions and/or increasing inflation. Government can also affect net exports. A gold standard decreases policy options and is deflationary. As prices go down, people hoard money, which deepens and prolongs a recession. Countries recovered from the Great Depression in the order that they exited the gold standard.