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 $\Z$.

- In the spectral world, the fiber product $M:=\spec\FF_2\times_{\spec \mathbb{S}} \spec \FF_2 = \spec(\FF_2\otimes_\mathbb{S} \FF_2)$ lives over $\spec \FF_2$, and $\pi_\ast \mathcal{O}_M$ can be identified with $\pi_\ast(\FF_2\otimes_\mathbb{S} \FF_2)$, which is the dual Steenrod algebra.
- In contrast, the fiber product $N:=\spec \FF_2\times_{\spec \Z} \spec \FF_2 = \spec(\FF_2\otimes^\mathbb{L}_\Z \FF_2)$ in the world of derived algebraic geometry is not the dual Steenrod algebra! Instead, $\pi_\ast \mathcal{O}_N$ can be identified with $\pi_\ast(\FF_2\otimes^\mathbb{L}_\Z \FF_2) = \mathrm{Tor}^\ast_\mathbf{Z}(\FF_2, \FF_2) = \FF_2[x]/x^2$ with $|x| = 1$.

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 $n>1$.