As Tyler pointed out, it is "too easy" to be representable in the non-connective world. This might sound good, but it comes at the cost of geometric intuition. It is related to the fact that negative homotopy groups of the cotangent complex arise from "stacky" phenomena, while in the non-connective setting it will be impossible to distinguish what comes from stackiness and what comes from non-connectiveness of the rings themselves. I will try to give an example of this below.

1. First, a slight reformulation of Tyler's example (just to show that this is a very general phenomenon). Let $X = Spec(A)$ be an affine scheme and $U \subset X$ a quasi-compact open subscheme.

Lemma: When considered as a nonconnective spectral scheme, $U$ is affine.

Proof: $U$ can be written as the vanishing locus of some perfect complex $F \in Perf(X)$. In other words, as a non-connective spectral stack, the functor of points of $U$ is as follows: a $T$-point $T \to U$ is a $T$-point $x : T \to X$ such that $x^*(F) = 0$. According to Prop. 1.2.10.1 in Toën–Vezzosi's HAG II, there exists a canonical epimorphism $A \to B$ of non-connective $E_\infty$-ring spectra such that $Spec(B)$ has the functor of points described. (This $B$ is discrete if and only if $U$ is actually affine as a classical scheme.)

2. Let $X$ be a (connective) spectral scheme and let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. Consider the relative Zariski spectrum, the (connective) spectral stack $Spec_X(\mathcal{A})$ whose space of $T$-points is $Maps_{\mathcal{O}_T\text{-alg}}(x^*(\mathcal{A}), \mathcal{O}_T)$, for any $X$-scheme $x : T \to X$. In particular you can take $\mathcal{A} = Sym_{\mathcal{O}_X}(F)$ for any perfect complex $F$; let $V_X(F) := Spec_X(Sym_{\mathcal{O}_X}(F))$ denote the "generalized vector bundle" associated to $\mathcal{F}$.

One can compute (see Theorem 5.2 in Antieau-Gepner) the relative cotangent complex of $V_X(F)$ at any point $s : T \to V_X(F)$, for $x : T \to X$, as $x^*(F)$. You can read off a lot of information about $V_X(F)$ from the cotangent complex. Namely, say $F$ is of tor-amplitude $[a,b]$. If $a \ge 0$, i.e. $F$ is of positive tor-amplitude (hence connective), then $V_X(F)$ is representable by a (connective) spectral scheme (which is affine over $X$): by Zariski descent, you can assume $X$ is affine, and then $V_X(F) = Spec(\Gamma(X, Sym_{\mathcal{O}_X}(F)))$. If $a \le 0$, then $V_X(F)$ is a spectral $(-a)$-Artin stack. If further $b \le 0$ then $V_X(F)$ is smooth.

In the non-connective world, the same argument will apply to show that $V_X(F)$ is a non-connective spectral scheme even when $F$ is non-connective. In other words, by passing to the non-connective world, we allowed ourselves to replace stacks by "schemes", but on the other hand we lost something significant: it is not clear anymore what information we can read from the cotangent complex about the geometry of the "scheme".

As Tyler pointed out, it is "too easy" to be representable in the non-connective world. This might sound good, but it comes at the cost of geometric intuition. It is related to the fact that negative homotopy groups of the cotangent complex arise from "stacky" phenomena, while in the non-connective setting it will be impossible to distinguish what comes from stackiness and what comes from non-connectiveness of the rings themselves. I will try to give an example of this below.

1. First, a slight reformulation of Tyler's example (just to show that this is a very general phenomenon). Let $X = Spec(A)$ be an affine scheme and $U \subset X$ a quasi-compact open subscheme.

Lemma: When considered as a nonconnective spectral scheme, $U$ is affine.

Proof: $U$ can be written as the vanishing locus of some perfect complex $F \in Perf(X)$. In other words, as a non-connective spectral stack, the functor of points of $U$ is as follows: a $T$-point $T \to U$ is a $T$-point $x : T \to X$ such that $x^*(F) = 0$. According to Prop. 1.2.10.1 in Toën–Vezzosi's HAG II, there exists a canonical epimorphism $A \to B$ of non-connective $E_\infty$-ring spectra such that $Spec(B)$ has the functor of points described. (This $B$ is discrete if and only if $U$ is actually affine as a classical scheme.)

2. Let $X$ be a (connective) spectral scheme and let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. Consider the relative Zariski spectrum, the (connective) spectral stack $Spec_X(\mathcal{A})$ whose space of $T$-points is $Maps_{\mathcal{O}_T\text{-alg}}(x^*(\mathcal{A}), \mathcal{O}_T)$, for any $X$-scheme $x : T \to X$. In particular you can take $\mathcal{A} = Sym_{\mathcal{O}_X}(F)$ for any perfect complex $F$; let $V_X(F) := Spec_X(Sym_{\mathcal{O}_X}(F))$ denote the "generalized vector bundle" associated to $F$.

One can compute (see Theorem 5.2 in Antieau-Gepner) the relative cotangent complex of $V_X(F)$ at any point $s : T \to V_X(F)$, for an $X$-scheme $x : T \to X$, as $x^*(F)$. You can read off a lot of information about $V_X(F)$ from the cotangent complex. Namely, say $F$ is of tor-amplitude $[a,b]$ (I'm going to use homological grading). If $a \ge 0$, i.e. $F$ is of non-negative tor-amplitude (hence connective), then $V_X(F)$ is representable by a (connective) spectral scheme (which is affine over $X$): by Zariski descent, you can assume $X$ is affine, and then $V_X(F) = Spec(\Gamma(X, Sym_{\mathcal{O}_X}(F)))$. If $a \le 0$, then $V_X(F)$ is a spectral $(-a)$-Artin stack (this is what I meant about the cotangent complex controlling "stackiness"). If further $b \le 0$ then $V_X(F)$ is smooth.

That was the connective story. In the non-connective world, $V_X(F)$ will "automatically" become representable by a non-connective spectral scheme even when $F$ is non-connective. In other words, by passing to the non-connective world, we allowed ourselves to replace stacks by "schemes", but on the other hand we lost something significant: it is not clear anymore what information we can read from the cotangent complex about the geometry of the "scheme".

As Tyler pointed out, it is "too easy" to be representable in the non-connective world. This might sound good, but it comes at the cost of geometric intuition. It is related to the fact that negative homotopy groups of the cotangent complex arise from "stacky" phenomena, while in the non-connective setting it will be impossible to distinguish what comes from stackiness and what comes from non-connectiveness of the rings themselves. I will try to give an example of this below.

1. First, a slight reformulation of Tyler's example (just to show that this is a very general phenomenon). Let $X = Spec(A)$ be an affine scheme and $U \subset X$ a quasi-compact open subscheme.

Lemma: When considered as a nonconnective spectral scheme, $U$ is affine.

Proof: $U$ can be written as the vanishing locus of some perfect complex $F \in Perf(X)$. In other words, as a non-connective spectral stack, the functor of points of $U$ is as follows: a $T$-point $T \to U$ is a $T$-point $x : T \to X$ such that $x^*(F) = 0$. According to Prop. 1.2.10.1 in Toën–Vezzosi's HAG II, there exists a canonical epimorphism $A \to B$ of non-connective $E_\infty$-ring spectra such that $Spec(B)$ has the functor of points described. (This $B$ is discrete if and only if $U$ is actually affine as a classical scheme.)

2. Let $X$ be a (connective) spectral scheme and let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. Consider the relative Zariski spectrum, the (connective) spectral stack $Spec_X(\mathcal{A})$ whose space of $T$-points is $Maps_{\mathcal{O}_T\text{-alg}}(x^*(\mathcal{A}), \mathcal{O}_T)$, for any $X$-scheme $x : T \to X$. In particular you can take $\mathcal{A} = Sym_{\mathcal{O}_X}(F)$ for any perfect complex $F$; let $V_X(F) := Spec_X(Sym_{\mathcal{O}_X}(F))$ denote the "generalized vector bundle" associated to $\mathcal{F}$.

One can compute (see Theorem 5.2 in Antieau-Gepner) the relative cotangent complex of $V_X(F)$ at any point $s : T \to V_X(F)$, for $x : T \to X$, as $x^*(F)$. You can read off a lot of information about $V_X(F)$ from the cotangent complex. Namely, say $F$ is of tor-amplitude $[a,b]$. If $a \ge 0$, i.e. $F$ is of positive tor-amplitude (hence connective), then $V_X(F)$ is representable by a (connective) spectral scheme (which is affine over $X$): by Zariski descent, you can assume $X$ is affine, and then $V_X(F) = Spec(\Gamma(X, Sym_{\mathcal{O}_X}(F)))$. If $a \le 0$, then $V_X(F)$ is a spectral $(-a)$-Artin stack. If further $b \le 0$ then $V_X(F)$ is smooth.

In the non-connective world, the same argument will apply to show that $V_X(F)$ is a non-connective spectral scheme even when $F$ is non-connective. In other words, by passing to the non-connective world, we allowed ourselves to replace stacks by "schemes", but on the other hand we lost something significant: it is not clear anymore what information we can read from the cotangent complex.

As Tyler pointed out, it is "too easy" to be representable in the non-connective world. This might sound good, but it comes at the cost of geometric intuition. It is related to the fact that negative homotopy groups of the cotangent complex arise from "stacky" phenomena, while in the non-connective setting it will be impossible to distinguish what comes from stackiness and what comes from non-connectiveness of the rings themselves. I will try to give an example of this below.

1. First, a slight reformulation of Tyler's example (just to show that this is a very general phenomenon). Let $X = Spec(A)$ be an affine scheme and $U \subset X$ a quasi-compact open subscheme.

Lemma: When considered as a nonconnective spectral scheme, $U$ is affine.

Proof: $U$ can be written as the vanishing locus of some perfect complex $F \in Perf(X)$. In other words, as a non-connective spectral stack, the functor of points of $U$ is as follows: a $T$-point $T \to U$ is a $T$-point $x : T \to X$ such that $x^*(F) = 0$. According to Prop. 1.2.10.1 in Toën–Vezzosi's HAG II, there exists a canonical epimorphism $A \to B$ of non-connective $E_\infty$-ring spectra such that $Spec(B)$ has the functor of points described. (This $B$ is discrete if and only if $U$ is actually affine as a classical scheme.)

2. Let $X$ be a (connective) spectral scheme and let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. Consider the relative Zariski spectrum, the (connective) spectral stack $Spec_X(\mathcal{A})$ whose space of $T$-points is $Maps_{\mathcal{O}_T\text{-alg}}(x^*(\mathcal{A}), \mathcal{O}_T)$, for any $X$-scheme $x : T \to X$. In particular you can take $\mathcal{A} = Sym_{\mathcal{O}_X}(F)$ for any perfect complex $F$; let $V_X(F) := Spec_X(Sym_{\mathcal{O}_X}(F))$ denote the "generalized vector bundle" associated to $\mathcal{F}$.

One can compute (see Theorem 5.2 in Antieau-Gepner) the relative cotangent complex of $V_X(F)$ at any point $s : T \to V_X(F)$, for $x : T \to X$, as $x^*(F)$. You can read off a lot of information about $V_X(F)$ from the cotangent complex. Namely, say $F$ is of tor-amplitude $[a,b]$. If $a \ge 0$, i.e. $F$ is of positive tor-amplitude (hence connective), then $V_X(F)$ is representable by a (connective) spectral scheme (which is affine over $X$): by Zariski descent, you can assume $X$ is affine, and then $V_X(F) = Spec(\Gamma(X, Sym_{\mathcal{O}_X}(F)))$. If $a \le 0$, then $V_X(F)$ is a spectral $(-a)$-Artin stack. If further $b \le 0$ then $V_X(F)$ is smooth.

In the non-connective world, the same argument will apply to show that $V_X(F)$ is a non-connective spectral scheme even when $F$ is non-connective. In other words, by passing to the non-connective world, we allowed ourselves to replace stacks by "schemes", but on the other hand we lost something significant: it is not clear anymore what information we can read from the cotangent complex about the geometry of the "scheme".