It doesn't seem to me that there were any substantive differences between the Bretton Woods regime and the gold standard, as seen in the early 1970s
Note that most of what I'm about to write is summarized from Wikipedia.
The gold standard of some form has always existed in the US since the Coinage Act of 1792 with a few interruptions. In 1934, gold was nationalized in the US but the value of the US dollar was still tied to gold reserves. Bretton Woods simply maintained the gold standard. While the rate between the US dollar and gold varied over the years, until 1976, the gold standard was officially kept. In other words, indeed there were no substantive differences because the Bretton Woods era was just a short part of the gold standard history.
I'd like to know why economists thought Bretton Woods was a good idea.
In the sense that they mostly kept policy that had been in existence for over a century by that point, it wasn't too surprising. The general summarized critique against the gold standard based on my reading of the IGM survey is that gold is too volatile compared to the existing system with the US dollar and the Fed and gold doesn't allow for any counter-cyclical monetary policy. If economists a century ago took the same survey it isn't clear that there would be such a clear consensus against the gold standard. As the St.Louis Fed writes,
For the pre-Fed period (1790-1913), the average annual inflation was
0.4 percent with a coefficient of variation of 13.2. During the period 1941-2016, these figures changed to 3.5 percent and 0.8, respectively.
If we look at the post-Volcker era (1988-2016), annual inflation was
2.2 percent on average with a coefficient of variation of 0.4.
In other words, we've come a long way in terms of keeping price volatility of everyday goods and services in control. Relative to the past, in developed countries, we are no longer concerned with hyperinflation, which a gold standard makes fairly difficult.
Also from the perspective of the US, it helped solidify the US dollar as a reserve currency. All of the benefits listed in this SE answer apply besides seigniorage.