In this comment it was asserted that Divide[a,b] and a/b are different, though the documentation indicates that they are the same. In particular, it was asserted that a/b is evaluated as a * 1/b, whereas Divide[a,b] performs the division directly. It was further asserted that this could result in an observable difference in behavior for machine-precision numbers.

The question is: is there a difference? If so, what is it exactly? How does it manifest itself? Is there an example where Divide gives a different result than /?

I tried a few edge cases with $MaxMachineNumber and $MinMachineNumber, but didn't find any differences in behavior between the two.