Hi,
there is not bug in the Wolfram|Alpha. But your expected result is not a general result; i.e you make additional assumptions. If you assume that a is a real number that is positive then the term simplifies to yours. But if a<0 for example Wolfram|Alpha's solution is more appropriate.
If you go to one of the (free) cloud products (not Wolfram|alpha) and type in:
FullSimplify[(2 a - (4 a^2)^(1/3))/(Sqrt[2 a] - (2 a)^(1/3)), Assumptions -> a \[Element] Reals && a > 0 ]
you will get the result you expect. For example:
Simplify[2^(1/3) a^(1/3) + Sqrt[2] Sqrt[a] == (2 a - (4 a^2)^(1/3))/(Sqrt[2 a] - (2 a)^(1/3)),Assumptions -> a \[Element] Reals && a > 0 ]
evaluates to True, whereas
Simplify[2^(1/3) a^(1/3) + Sqrt[2] Sqrt[a] == (2 a - (4 a^2)^(1/3))/(Sqrt[2 a] - (2 a)^(1/3))]
evaluates to
(a^(2/3) - (a^2)^(1/3))/((-1 + 2^(1/6) a^(1/6)) a^(1/6)) == 0
Cheers,
Marco