I did some related work on human mobility these days and came across the terms of Power-Law, Pareto, Zipf’s and Scale-Free distributions all the time. And, shame on me, I did not know the “difference”. Indeed, it turned out that all these notions are words for the same thing as explained by
Power laws, Pareto distributions and Zipf’s law
Contemporary physics46 (5): 323-351 (2005)
In particular it says about the power-law, Zipf’s law and the Pareto distribution:
Since power-law cumulative distributions imply a powerlaw form for p(x), “Zipf’s law” and “Pareto distribution” are effectively synonymous with “power-law distribution”.
With regard to the scale-free aspect of power-law, it says:
A power-law distribution is also sometimes called a scale-free distribution. Why? Because a power law is the only distribution that is the same whatever scale we look at it on.
And, finally, a little fun fact:
Zipf’s law and the Pareto distribution differ from one another in the way the cumulative distribution is plotted—Zipf made his plots with x on the horizontal axis and P(x) on the vertical one; Pareto did it the other way around. This causes much confusion in the literature, but the data depicted in the plots are of course identical.