"depending on whether" is indeed the most commo0n way to use "depending" to indicate a choice of two alternatives. But the sentence can often be improved by rewriting to specify the alternatives more exactly, which often involves not using "depending" at all.
For instance, take the sentence given in the answer by Lambie.
He uses a car depending on whether his meetings are in another city or in his own city.
Does that mean "He uses a car for meetings in his own city, but travels by air to meetings in other cities" or does it mean, say "He walks or takes the subway to meetings in his own city, but drives to meetings in other cities"? One cannot tell from the "depending on whether" sentence. Specifying exactly what result follows which condition gives the reader more information. That form also usually does away with any use of "depending".
"Depending upon" or "depending on" can be used to indicate an action followed only in a particular circumstance. For example
I may get bids from several vendors, depending upon the amount of the transaction.
This implies that for a larger transaction, multiple bids are sought, but not for a small one.
I may have Chinese or Mexican for lunch, depending on my mood and how busy I am.
This suggests but does not say that one choice takes longer, and so is not used when i am busy. But it doesn't say which one. Perhaps the listener is supposed to know, or not to care. Or perhaps a later sentence makes this clear.
There is little or no difference in meaning between "depending on" and "depending upon". The second is a bit more formal. Both tend to state a condition where part of the relation is implied, and must be known from the logic of the situation or from context.