Is there any difference between these two phrases?
For example, are they interchangble in these two sentences:
I've been putting money aside to buy a piano.
The international body put aside October 1st to tackle the new outbreak of Ebola.
No substantial difference between the two. They refer to exactly the same action.
There is no difference in the phrases "put aside" and "set aside" when they are used to save money or time or keep a period of time free in order to use later for a particular purpose.
These phrases are interchangeable in the sentences mentioned in the question.
For me, "put" is open to interpretation of how to carry out the action.
"Put your things on the counter." (could be dropped, tossed or thrown. The outcome is the only thing that matters)
On the other hand, "set" has a gentle, elegant or careful meaning to it. (e.g. "Set the table")
"Set your things on the counter." (The actor takes care when placing the things)
So how does this fit with "Set aside" vs. "Put aside"?
"Set aside $100" would imply care, importance, and necessity.
"Put aside $100" would imply easy, simple, obvious.