I don't know a really satisfying answer to this question, but here are a few observations.
1) The $\infty$-category of simplicial commutative $k$-algebras is monadic over the $\infty$-category of connective $k$-module spectra. The relevant monad is the nonabelian left derived functor of the "total symmetric power" on ordinary $k$-modules, which is different from
the construction $M \mapsto \bigoplus_{n} (M^{\otimes n})_{h \Sigma_n}$ unless $k$ has characteristic zero.
2) The $\infty$-category of simplicial commmutative rings is freely generated under sifted colimits by the ordinary category of finitely generated polynomial algebras over $k$.
In other words, it can be realized as the $\infty$-category of product-preserving functors from the ordinary category of $k$-schemes which are affine spaces to the $\infty$-category of spaces.
3) Let $X$ be the affine line over $k$ (in the sense of classical algebraic geometry). Then
$X$ represents the forgetful functor {commutative $k$-algebras} -> {sets}. Consequently,
$X$ has the structure of a commutative $k$-algebra in the category of schemes.
Also, $X$ is flat over $k$.
Now, any ordinary scheme can be regarded as a spectral scheme over $k$: that is, it also represents a functor {connective E-infty algebras over k} -> {spaces}. In general, products in the category of ordinary $k$-schemes need not coincide with products in the $\infty$-category of spectral $k$-schemes. However, they do agree for flat $k$-schemes. Consequently, $X$ can be regarded as a commutative $k$-algebra in the $\infty$-category of derived $k$-schemes. In particular, $X$ represents a functor {connective E-infty algebras over k} -> {connective E_infty algebras over k}. This functor has the structure of a comonad whose comodules are the simplicial commutative $k$-algebras.
You can summarize the situation more informally by saying: derived algebraic geometry (based on simplicial commutative $k$-algebras) is what you get when you take
spectral algebraic geometry (based on E-infty-algebras over $k$) by forcing the two different versions of the affine line to coincide.
4) The forgetful functor {simplicial commutative $k$-algebras} -> {E-infty algebras over $k$} is both monadic and comonadic. In particular, you can think of a simplicial commutative $k$-algebra $R$ as an E-infty algebra over $k$ with some additional structure. As Tyler mentioned, one way of thinking about that additional structure is that it gives you the ability to form symmetric powers of connective modules. Of course, if $M$ is any $R$-module spectrum, you can always form the construction $(M^{\otimes n})_{h \Sigma_n}$. However, this doesn't behave the way you might expect based on experience in ordinary commutative algebra: for example, if $M$ is free (i.e. a sum of copies of $R$) then $(M^{\otimes n})_{h \Sigma_n}$ need not be free (unless $R$ is of characteristic zero). However, when
$R$ is a simplicial commutative ring, there is a related construction on connective
$R$-module spectra, given by nonabelian left derived functors of the usual symmetric power.
This will carry free $R$-module spectra to free $R$-module spectra (of the expected rank).
It is possible to describe the $\infty$-category of simplicial commutative $k$-algebras along the following lines: a simplicial commutative $k$-algebra is
a connective E-infty algebra over $R$ together with a collection of symmetric power functors
Sym^{n} from connective $R$-modules to itself, plus a bunch of axioms and coherence data.
I don't remember the exact statement (my recollection is that spelling this out turned out to be more trouble than it was worth).
In characteristic zero, the model structure on commutative dg-algebras is obtained by transfer from the projective model structure on chain complexes, along the ajunction between the free algebra functor and the forgetful functor. In particular, weak equivalences and fibrations are determined in chain complexes (quasi-isomorphims and degreewise surjections).
In positive characteristic, a model structure still exists, which is actually available for commutative dg-algebras over any commutative ring: this was proved by Don Stanley in his preprint Determining closed model category structures.
However, this model structure is not nice, in the sense that fibrations are not necessarily surjective in positive degrees: weak equivalences and fibrations are not determined by the forgetful functor from commutative dg-algebras to chain complexes.
Actually this is an incarnation of a more general fact about the possibility to transfer a model structure from a model category to its category of commutative monoids. A nice criterion for this is called the commutative monoid axiom in the paper of David White Model structures on commutative monoids in general model categories, and it turns out that such a criterion fails for commutative dg-algebras in positive characteristic.
Now, going back to derived geometry, a good model that works in positive characteristic is the one of simplicial rings, which inherits a nice model structure in any characteristic. Moreover, in characteristic zero, simplicial rings are Quillen equivalent to commutative dg-algebras (equipped with the model structure induced by the one of chain complexes) via the Dold-Kan correspondence.
I would like also to point out that, in characteristic zero, using commutative dg-algebras as affine derived stacks can be very useful to, for instance, study geometric structures on derived stacks such as shifted symplectic structures. The paper Shifted symplectic structures by Pantev-Toen-Vaquié-Vezzosi is written in this context.
Best Answer
I'll try to answer this question from the topological viewpoint. The short summary is that structured objects in the spectral setting have cohomology operations and power operations, which forces spectral algebraic geometry to be different from derived algebraic geometry.$\newcommand{\FF}{\mathbf{F}}\DeclareMathOperator{spec}{Spec}\newcommand{\Eoo}{E_\infty} \newcommand{\Z}{\mathbf{Z}} \newcommand{\GL}{\mathrm{GL}} \newcommand{\N}{\mathbf{N}} $
Let us illustrate this by a few examples. The simplest example is that in positive characteristic, any discrete ring $A$ has a Frobenius endomorphism. Simplicial commmutative rings are freely generated under sifted colimits by finitely generated polynomial $A$-algebras, so in positive characteristic, simplicial commutative rings have Frobenius endomorphisms. However, in general, $\Eoo$-rings have nothing like a Frobenius endomorphism.
For a few more examples, consider the discrete ring $\FF_2$, regarded as an $\Eoo$-algebra over the sphere, and as a discrete simplicial commutative ring, as in my comment. Note that the initial object in the category of simplicial commutative rings is the usual ring $\Z$.
This is already one example of how the spectral and derived worlds diverge in positive characteristic. Another example comes from considering the affine line. In the classical world, the affine line $\mathbf{A}^1_X$ is flat over the discrete scheme $X$. Taking the functor of points approach, the affine line is defined as $\spec$ of the free commutative algebra on one generator (i.e., the polynomial ring $\Z[x]$).
In the spectral world, the free $\Eoo$-algebra on one generator over an $\Eoo$-ring $R$ is given by $\bigoplus_{n\geq 0} (R^{\otimes n})_{h\Sigma_n} =: R\{x\}$ (see the note at the end of this answer). For instance, this means that $\mathbb{S}\{x\}$ is a direct sum of the spectra $\Sigma^\infty B\Sigma_n$. Suppose that $R = \FF_2$, again regarded as a discrete $\Eoo$-ring. Then $$\pi_n\FF_2\{x\} = \bigoplus_{k\geq 0} H_n(\Sigma_k; \FF_2),$$ where each $\Sigma_k$ acts trivially on $\FF_2$. In particular, $\FF_2\{x\}$ is not flat over $\FF_2$. (Recall that if $A$ is an $\Eoo$-ring, then an $A$-module $M$ said to be flat over $A$ if $\pi_0 M$ is flat over $\pi_0 A$ and the natural map $\pi_\ast A\otimes_{\pi_0 A} \pi_0 M\to \pi_\ast M$ is an isomorphism. In particular, any flat module over a discrete ring must be concentrated in degree zero.) This failure corresponds exactly to the existence of Steenrod operations. More generally, the homotopy groups $\pi_\ast R\{x\}$ carry all information about power operations on $E_\infty$-$R$-algebras.
On the other hand, the free simplicial commutative ring on one generator over $\FF_2$ is just the polynomial ring $\FF_2[x]$. This is certainly flat over $\FF_2$.
Yet another example (technically not "positive characteristic", but is still a good example illustrating the difference between the spectral and derived worlds) along these lines comes from contemplating the definition of the scheme $\mathbf{G}_m$. Classically, this is defined as $\spec$ of the free commutative algebra on one invertible generator.
In the spectral world, a natural candidate for this functor already exists: it is the functor known as $\GL_1$. If $R$ is an $\Eoo$-ring, one can define the space $\GL_1(R)$ as the pullback $\Omega^\infty R \times_{\pi_0 R} (\pi_0 R)^\times$, i.e., as the component of $\Omega^\infty R$ lying over the invertible elements in $\pi_0 R$. (One can similarly define $\mathrm{SL}_1(R)$.) We run into the same problem --- the resulting spectral scheme $\GL_1$ is not flat over the sphere spectrum. The space $\GL_1(R)$ is an infinite loop space, and is very mysterious in general.
In the derived world, one can define the functor $\mathbf{G}_m$ as $\spec$ of the free simplicial commutative ring on one invertible generator, i.e., as $\spec \Z[x^{\pm 1}]$. In fact, one can then extend the input of $\mathbf{G}_m$ to $\Eoo$-rings, by defining $\mathbf{G}_m(A)$, for $A$ an $\Eoo$-ring, to be $\mathrm{Map}_{\text{infinite loop}}(\Z, \GL_1(A))$. Then, there is a map of schemes $\mathbf{G}_m\to \GL_1$ which is an equivalence over the rationals.
This point of view (that cohomology operations and power operations separate the spectral and derived worlds) also helps ground one's intuition for why simplicial commutative rings and $\Eoo$-rings agree rationally: the rational Steenrod algebra is trivial (the rational sphere is just $\mathbf{Q}$)! Of course, this doesn't comprise a proof.
Note that in Lurie's books, the affine line is not defined via the free $\Eoo$-ring on one generator; as we saw above, that $\Eoo$-ring is a bit unwieldy. Instead, $\mathbf{A}^1_R$ is defined to be $\spec(R\otimes_\mathbb{S} \Sigma^\infty_+ \mathbf{N})$, where $\mathbf{N}$ is now regarded as a discrete topological space. As $\Sigma^\infty_+ \mathbf{N}$ is flat over $\mathbf{S}$, this sidesteps the non-flatness problem mentioned above. Replacing $\N$ with $\Z$ above, one similarly sidesteps the non-flatness issue for $\mathbf{G}_m$.
I want to comment briefly on a surprising example where the spectral and derived worlds agree. Instead of studying spectral algebraic geometry with $\Eoo$-rings, let us work in the context of spectral algebraic geometry with $E_2$-rings. (In the hierarchy of $E_k$-rings, this is the minimum structure one can/should impose before it is reasonable to say that there is some level of commutativity. For instance, if $A$ is an $E_k$-ring, then $\pi_0 A$ is a commutative ring once $k\geq 2$.) The theory of algebraic geometry over $E_k$-rings was studied by Francis in his thesis (see http://www.math.northwestern.edu/~jnkf/writ/thezrev.pdf).
In the derived world, the free commutative algebra with $p=0$ is just $\FF_p$. Surprisingly, the free ($p$-local) $E_2$-ring with $p=0$ is also $\FF_p$! This result is due to Hopkins and Mahowald. Other than the actual proof (which is very enlightening, and a nice application of $E_2$-Dyer-Lashof operations; see https://arxiv.org/abs/1411.7988 and https://arxiv.org/abs/1403.2023), I don't have any conceptual explanation for why this result should morally be true. Note, however, that the free ($p$-local) $E_2$-ring with $p^n=0$ is not discrete (this is due to Jeremy Hahn, see https://arxiv.org/abs/1707.00956) for $n>1$.