I am a little confused about why two different approaches are giving two different answers.
The most natural approach for me is to consider the dice ordered at first, so we have 6 options for the first dice, 6 for the second ... and so on so we get $$6^9$$ But since it is unordered we don't care which dice is the first we can mod out by $9!$ giving $$\frac{6^9}{9!}$$ But we can also solve it using the stars and bars method (allowing 0s for any of the buckets) so we get $$\binom{14}5$$ Why are these different?
$\frac{6^9}{9!}$ does not give the correct answer for any situation, because some ordered combinations of dice rolls remain invariant under certain permutations – for example, any roll where all dice show the same number remains invariant under any permutation. Therefore $9!$ is not the correct factor to divide by.
$\binom{14}5$ is the correct answer.