pr.probability - Sample from uniform distribution vs. Sample from random distribution

You're right that things change for m>1; I was thinking sloppily.

Assume $U={1,ldots,n}$ for concreteness. If $Y_1,ldots,Y_m$ are chosen independently and uniformly from $U$, then for any $k_1,ldots,k_min U$, we of course have
$$
Pr[Y_1=k_1,ldots,Y_m=k_m] = frac{1}{n^m}.
$$

On the other hand, if $x=(x_1,ldots,x_m)$ is chosen uniformly from the standard $n$-simplex and $Y_1,ldots,Y_m$ are then chosen independently according to $x$, then
$$
Pr[Y_1=k_1,ldots,Y_m=k_m] = mathbb{E}Pr[Y_1=k_1,ldots,Y_m=k_m|x]
= mathbb{E}prod_{i=1}^m x_{k_i} = frac{n!}{(n+r)!}prod_{j=1}^n r_j!,
$$
where $r_j = #{1le i le m : k_i=j}$ and $r=r_1 + cdots r_n$. This last expectation can be proved most easily from Lemma 1 in this paper.