The Fibonacci ratio is defined by the following sequence (there's more to it, but simplified):

Essentially take the prior two numbers and add to get the next.

The golden ratio is related that it expresses the ideal mathematical ratio between these numbers.
It is defined as 0.61803398874989

The Fibonacci does not represent this value until you get very far down the sequence, as it starts out at 1, then .5, then .66666667 etc...

As you may already know, Fibonacci represents how it occurs in real life, a sea shell for instance and how it grows.

The Golden Ratio is exact.
So when calculating harmonic changes when the frequency limits are in place, to achieve the accuracy of the golden ratio, one will have to go through many harmonic shifts. Typically a frequency will only need from 1 to 8 shifts to get within range.

Even at 8 harmonic shifts, the ratio is only 0.61818181818181818181818181818182, close but not the final true ratio.

So select the one that best suits your goals.

In other terms,
When you select Fibonacci, it will take the frequency and add it to itself, then add that to the prior frequency, and use this method to calculate what frequency to use when shifting up to a higher harmonic.

When you select Golden Ratio, it will take the frequency and add the result of multiplying the frequency by 0.61803398874989 to use when shifting up to a higher harmonic.

Reverse this process when bringing a frequency down to within limits.

Just to be a bit more complete, I gave the # for the ratio, but the definition for the golden ratio is as follows:
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.