I'll give an example, if you look in the docstring , 
>>> egypt_fraction(Rational(3, 7), "Greedy")
[3, 11, 231]
>>> egypt_fraction(Rational(3, 7), "Graham Jewett")
[7, 8, 9, 56, 57, 72, 3192]
>>> egypt_fraction(Rational(3, 7), "Takenouchi")
[4, 28, 7]

Here the input Rational(3,7) is same, but algos are different .. this is the difference i'm taking about ..