I have nothing non-trivial and non-digressive to say, but it might help to
point out in an elementary way some things that may be relevant. One way to think
about things is that there are distinctions in algebra that lack equivalents
in spectra. This relates to answers from the $\infty$ category point
of view of the original question, but is perhaps more explicit and concrete.

The statement of the question might be a little confusing,
since there are perhaps six rather than two natural categories
to consider, so let me pedantically make a fuller list, with
everything over some commutative ring $k$. As a joke start, notice
that simplicial commutative algebras are the same as commutative
simplicial algebras. However, simplicial $E_{\infty}$ algebras
make no sense (since we are not thinking about a homotopy theory
on plain algebras), whereas $E_{\infty}$ simplicial algebras might
make sense: as I understand it, that is where the problem about
symmetric powers enters.

(1) commutative DG $k$-algebras

(2) $E_{\infty}$ DG $k$-algebras

(3) commutative simplicial $k$-algebras = simplicial commutative $k$-algebras

(4) $E_{\infty}$ simplicial $k$-algebras?

(5) commutative $Hk$-algebras in any good category of spectra.

(6) $E_{\infty}$ $Hk$-algebras in any good category of spectra

The term $k$-module spectra in the question and some answers means $Hk$-module,
I presume, where $Hk$ is the Eilenberg-MacLane spectrum for $k$. I find it helpful
to maintain a notational distinction. Mandell proved that $E_{\infty}$ $k$-algebras
(algebra) are equivalent to $E_{\infty}$ $Hk$-algebras (topology). That is,
(2) and (6) are equivalent: the algebraic and topological notions of $E_{\infty}$
are equivalent, of course restricting the latter to Eilenberg-MacLane algebras.

In algebra, the evident forgetful functor from commutative DG $k$-algebras to
$E_{\infty}$ $k$-algebras is an equivalence for fields of characteristic 0
but not in general otherwise. That is, (1) and (2) are not equivalent.

Analogously, if sense can be made of (4), then there is a functor from commutative simplicial
$k$-algebras to $E_{\infty}$ simplicial $k$-algebras which is not an equivalence.

In topology, the world of spectra, the smash product builds in higher homotopies
and there is no distinction between commutative $Hk$-algebras and $E_{\infty}$
$Hk$-algebras: (5) and (6) are equivalent. Andre's answer is an $\infty$-categorical
version of this: "in general, there is nothing more that $E_{\infty}$ to ask for".
But the real topological reason in the case of spectra is a miracle of the modern theory
of spectra. For good spectra in any good category of spectra, the natural map from
the homotopy symmetric power to the symmetric power

`\[ (E\Sigma_n)_{+}\wedge_{\Sigma_n} X^n \longrightarrow X^n/\Sigma_n \]`

is an equivalence, where $X^n$ is the $n$-fold smash power. (I think I've advertised
this in answer to some other question, but it is worth repeating.) One cannot expect
anything like this in algebra, except when working over a field of characteristic $0$.

In the model categorical sense, there does not seem to be a good homotopy
theory of commutative DG $k$-algebras (ignoring the trivial rational case), and by
analogy one does not expect such a good homotopy theory of simplicial commutative
algebras. There is a good model theoretic homotopy theory of $E_{\infty}$ DG-algebras.

In the non-commutative case, the question has been answered, mainly by Brooke Shipley.

With the word commutative deleted and $E_{\infty}$ replaced by $A_{\infty}$, I'm pretty
sure that all 6 homotopy categories make sense and are equivalent.

Several people have mentioned equivariant homotopy theory. Akhil, the cartesian
power (or smash power in the based context) $X^n$ of a space $X$ is clearly a
$\Sigma_n$-space; nothing to that. Maybe you are thinking of genuine spectra.
Either way, Elmendorf's theorem is not particularly relevant. The earlier question
you referred to asked for a homotopy colimit version of the infinite symmetric power
of spaces. The equivalence above can be jacked up to say that in modern categories
of spectra, the naive homotopy colimit and categorical colimit construction of infinite
symmetric powers of spectra give equivalent answers, which is bizarre from a pre-1990s
view of homotopy theory. There are interesting old-fashioned symmetric powers of spectra,
and they are different. I don't understand them in the modern world.

In the world of $G$-spectra, things are much more interesting. There are many types of
operadically defined commutative ring $G$-spectra. This is new work starting with Hill and Hopkins
and continued by Blumberg and Hill, as yet not written down (at least not yet for public consumption). The comparison between algebra and topology here is fascinating and will be relevant to representation theory as well as to algebraic topology, or so I think. The old work by Lewis and myself is that part of the story that most directly mimics the nonequivariant case. I think I see how to understand equivariant units in the new context, and there is a relevant paper by Santhanam in the "classical" context, but there is more to be said even there, part of a thorough treatment of equivariant infinite loop space theory now in progress at Chicago.