I think that the issue is that the two expressions are fundamentally different in terms of how they are evaluated. Though they may, in some sense have the same amount of nesting (but actually they don't), that is not the issue with regard to when the branch cut comes into play.

Remember that Mathematica evaluates its expressions (in most cases) from the inside out. So in the case of

1/((x^2 - 1)^4)^(1/5)

a value of x between -1 and 1 is first raised to the 4th power before it (now a positive number) is then raised to the 1/5 power. Thus the branch cut is not encountered. But in the same range

1/(x^2 - 1)^(4/5)

a negative number is directly being raised to the 4/5 power.

This is presumably due to the fact that the Power function (which is what is being used in the form of the expression in the second example) is defined with a particular convention for a branch cut. From the documentation for Power:

Power[x,y] has a branch cut discontinuity for non-integer y running from -[Infinity] to 0 in the complex x plane.

FullForm[1/((x^2 - 1)^4)^(1/5)]

gives

Power[Power[Plus[-1,Power[x,2]],4],Rational[-1,5]]

whereas

FullForm[1/(x^2 - 1)^(4/5)]

gives

Power[Plus[-1, Power[x, 2]], Rational[-4, 5]]