Homology of spectra vs homology of infinite loop spaces

$\newcommand{\H}{\mathrm{H}} \newcommand{\Z}{\mathbf{Z}}$Let $X$ be a space. Then the $E$-(co)homology of $X$ is the same as the $E$-(co)homology of its suspension spectrum, i.e., $E_\ast(X) \cong E_\ast(\Sigma^\infty_+ X)$ (and you remove the basepoint in $\Sigma^\infty_+ X$ to get reduced $E$-homology). In the case when $E = \H\Z$, this says that $\H_\ast(X;\Z) \cong \H_\ast(\Sigma^\infty_+ X;\Z)$. In the case when $E$ is the sphere spectrum, this says that the stable homotopy groups of $X$, i.e., the unstable homotopy groups of $QX$, are the same as the homotopy groups of the spectrum $\Sigma^\infty X$.

Let's now turn to the homology of $QX$. We know that $\H_\ast(QX;\Z) \cong \H_\ast(\Sigma^\infty_+ QX;\Z)$, so you need to understand $\Sigma^\infty_+ QX$. A theorem of Snaith's says that there is an equivalence $$\Sigma^\infty_+ QX \simeq \bigvee_{n\geq 0} (\Sigma^\infty X)^{\wedge n}_{h\Sigma_n},$$ so we find that $$\H_\ast(QX;\Z)\cong \bigvee_{n\geq 0} \H_\ast((\Sigma^\infty X)^{\wedge n}_{h\Sigma_n};\Z).$$ The groups $\H_\ast((\Sigma^\infty X)^{\wedge n}_{h\Sigma_n};\Z)$ can be computed by a homotopy orbits spectral sequence: $$E_2^{s,t} = \H_\ast(\Sigma_n; \H_\ast((\Sigma^\infty X)^{\wedge n};\Z)) \Rightarrow \H_\ast((\Sigma^\infty X)^{\wedge n}_{h\Sigma_n};\Z).$$ If $\Z$ is replaced by $\mathbf{Q}$, then this spectral sequence degenerates (the $E_2$-page vanishes for $s>0$), and you find that $\H_\ast((\Sigma^\infty X)^{\wedge n}_{h\Sigma_n};\mathbf{Q}) \cong \H_\ast(\Sigma^\infty X;\mathbf{Q})^{\otimes n}_{\Sigma_n}$, i.e., $\H_\ast(QX;\mathbf{Q}) \cong \mathrm{Sym}^\ast \H_\ast(\Sigma^\infty X;\mathbf{Q})$. By the way, something special happens with $\mathbf{Q}$-cohomology of spectra: since the $\mathbf{Q}$-localization of the sphere spectrum is $\H\mathbf{Q}$, you find that $\pi_\ast(F_\mathbf{Q}) \cong \H_\ast(F_\mathbf{Q};\mathbf{Q}) \cong \H_\ast(F;\mathbf{Q})$ for any spectrum. (This is more generally true for any smashing localization.) (Edit: Whoops, I didn't actually finish my answer.) This means that $\H_\ast(\Sigma^\infty X;\mathbf{Q}) \cong \pi_\ast(\Sigma^\infty X_\mathbf{Q}) \cong \pi_\ast(\Sigma^\infty X) \otimes \mathbf{Q} = \pi_\ast^s(X) \otimes \mathbf{Q}$, so you find that $$\H_\ast(QX;\mathbf{Q}) \cong \mathrm{Sym}^\ast(\pi_\ast^s(X) \otimes \mathbf{Q}),$$ as you said in your post. The homology of $QX$ with coefficients in a general homology theory $E$ is a lot more complicated: if $E$ is a structured ring spectrum, it describes what are known as Dyer-Lashof operations for $E$.

Let us denote by $B^\infty : \infty\rm{LoopSpaces} \stackrel{\sim}{\to} \rm{ConnectiveSpectra}$ the equivalence of ($\infty$-)categories you hinted in your question. Let us first consider the case of reduced homology of a pointed CW-complex $X$, as formulas are slightly cleaner. Unwinding the definitions, we see that

$$\pi_n(\Sigma^\infty X \wedge H\mathbb Z) = \pi_n(B^\infty (QX) \wedge H\mathbb Z)$$ and $$\widetilde H_*(QX;\mathbb Z) = \pi_n(\Sigma^\infty (QX) \wedge H\mathbb Z) = \pi_n(B^\infty(QQX) \wedge H\mathbb Z)$$

$$H_*(QX;\mathbb Z) = \pi_n(\Sigma^\infty_+ (QX) \wedge H\mathbb Z) = \pi_n(B^\infty(Q((QX)_+)) \wedge H\mathbb Z).$$

So, essentially the difference lies in how many times you apply the $Q$ functor to $X$ before taking the homology of the associated connective spectrum.

When $X$ is unpointed, the fomulas above become $$\pi_n(\Sigma^\infty_+ X \wedge H\mathbb Z) = \pi_n(B^\infty (Q(X_+)) \wedge H\mathbb Z)$$ and $$\widetilde H_*(Q(X_+);\mathbb Z) = \pi_n(\Sigma^\infty (Q(X_+)) \wedge H\mathbb Z) = \pi_n(B^\infty(QQ(X_+)) \wedge H\mathbb Z)$$

$$H_*(Q(X_+);\mathbb Z) = \pi_n(\Sigma^\infty_+ (Q(X_+)) \wedge H\mathbb Z) = \pi_n(B^\infty(Q(Q(X_+)_+)) \wedge H\mathbb Z).$$

One way to clarify what happens is to keep track of which structure you want to consider on $QX$. By definition, the functor $Q$ carries pointed spaces to infinite loop spaces (that is, spaces with some extra structure). One can say that computing the homology of $X$, is equivalent to compute the homology of $QX$ as an infinite loop space (that is, the homology of the spectrum $B^\infty Q X$), whereas if you forget this extra structure on $QX$ and just look at its underlying pointed space, to know its homology you'll have to apply again the $Q$ functor (in order to land again in infinite loop spaces), and then compute its homology in the appropriate ($\infty$-)category.