Clebsch-Gordan coefficients: General Expression Does Not Match Specific Expression

An argument could be made that the result being returned by ClebschGordan[] is generically correct; that is, the expression that comes from the hypergeometric representation of the Clebsch-Gordan coefficient is correct except at a countable number of values.

In particular,

ClebschGordan[{2, 0}, {l2, 0}, {2, 0}] // FullSimplify
   Piecewise[{{(Sqrt[5] (-1 + l2) (2 + l2) (4 + l2) Sqrt[Gamma[5 - l2]/Gamma[6 + l2]])/
               ((-4 + l2) Gamma[3 - l2]), l2 ∈ Integers && 0 <= l2 <= 4}}, 0]

Looking at the expression inside the conditional,

Table[(Sqrt[5] (-1 + l2) (2 + l2) (4 + l2) Sqrt[Gamma[5 - l2]/Gamma[6 + l2]])/
      ((-4 + l2) Gamma[3 - l2]), {l2, 0, 4}]
   {1, 0, -Sqrt[2/7], 0, Indeterminate}

we do get the Indeterminate result noted in the OP, but you should also account for the following:

Limit[(Sqrt[5] (-1 + l2) (2 + l2) (4 + l2) Sqrt[Gamma[5 - l2]/Gamma[6 + l2]])/
      ((-4 + l2) Gamma[3 - l2]), l2 -> 4]
   Sqrt[2/7]

which is the expected answer. In short, the result is usually correct except at l2 = 4, and in that special case, a limit must be taken.