The Gelbart-Jacquet lift of a $\mathrm{GL}_2$ cuspidal automorphic representation $\pi$ is an automorphic representation $\Pi$ of $\mathrm{GL}_3$; it is the adjoint or symmetric square (well, not quite, but these are equal up to a twist).

  More precisely, if $\pi$ has central character $\omega_{\pi}$, then the symmetric square $L$-function is defined such that $L(s,\pi \otimes \pi) = L(s,\mathrm{sym}^2 \pi) L(s,\omega_{\pi})$, while the adjoint $L$-function is defined such that $L(s,\pi \otimes \widetilde{\pi}) = L(s,\mathrm{ad} \pi) \zeta(s)$. One can use the $\mathrm{GL}_3$ converse theorem to show that $L(s,\mathrm{sym}^2 \pi)$ and $L(s,\mathrm{ad} \pi)$ are both $L$-functions of automorphic representations of $\mathrm{GL}_3$.

(As an aside, note that these need not be cuspidal; see Poles of $L$-functions associated to Maass forms)

Gelbart-Jacquet lifts are rare among all $\mathrm{GL}_3$ automorphic representations.

The first way to quantify this is a result of Ramakrishnan; which states that there is (essentially) a bijection between Gelbart-Jacquet lifts and self-dual automorphic representations of $\mathrm{GL}_3$. Unlike for $\mathrm{GL}_2$, being self-dual is a very restrictive statement for $\mathrm{GL}_n$ with $n \geq 3$.

The second way to quantify this is via Weyl's law. A result of Guerreiro gives a Weyl law for Gelbart-Jacquet lifts with main term $\asymp T^4/\sqrt{\log T}$, whereas the Weyl law for all $\mathrm{GL}_3$ forms has main term $\asymp T^5$.

A third way of viewing this is to look locally. Let $\Pi$ be an automorphic representation of $\mathrm{GL}_3$. At almost all places $v$, the local component $\Pi_v$ is an unramified principal series representation $|\cdot|_v^{t_{v,1}} \boxplus |\cdot|_v^{t_{v,2}} \boxplus |\cdot|_v^{t_{v,3}}$, where the spectral parameters $t_{v,1},t_{v,2},t_{v,3}$ satisfy $t_{v,1} + t_{v,2} + t_{v,3} = 0$ and $|\Re(t_{v,j})| < 1/2$ (actually, I think the best known bounds towards Ramanujan in this case is $5/14$). Self-dual representations satisfy the additional (quite stringent!) restriction that $t_{v,j} = 0$ for some $j \in \{1,2,3\}$.

The Gelbart-Jacquet lift of a $\mathrm{GL}_2$ cuspidal automorphic representation $\pi$ is an automorphic representation $\Pi$ of $\mathrm{GL}_3$. More generally, the Gelbart-Jacquet lift $\Pi$ is a $\mathrm{GL}_3$ automorphic representation associated to a $\mathrm{GL}_2$ automorphic representation $\pi$ and a Hecke character $\chi$ such that $L(s,\pi \otimes \widetilde{\pi} \otimes \chi) = L(s,\Pi) L(s, \chi)$. one can use the $\mathrm{GL}_3$ converse theorem to show that $L(s,\Pi)$ is an $L$-function of an automorphic representation of $\mathrm{GL}_3$.

When $\chi = 1$, $\Pi$ is equal to $\mathrm{ad} \pi$, the adjoint. When $\chi = \omega_{\pi}^{-1}$, the inverse of the central character of $\pi$, so that $\widetilde{\pi} \otimes \chi = \pi$, $\Pi$ is equal to $\mathrm{sym}^2 \pi$. (In particular, these two are equal up to a twist.)

As an aside, note that the Gelbart-Jacquet lift need not be cuspidal; see Poles of $L$-functions associated to Maass forms.

Gelbart-Jacquet lifts are rare among all $\mathrm{GL}_3$ automorphic representations.

The first way to quantify this is a result of Ramakrishnan; which states that there is (essentially) a bijection (after twisting, if necessary) between Gelbart-Jacquet lifts and self-dual automorphic representations of $\mathrm{GL}_3$. Unlike for $\mathrm{GL}_2$, being self-dual is a very restrictive statement for $\mathrm{GL}_n$ with $n \geq 3$.

The second way to quantify this is via the Weyl law. A result of Guerreiro gives a Weyl law for Gelbart-Jacquet lifts with main term $\asymp T^4/\sqrt{\log T}$, whereas the Weyl law for all $\mathrm{GL}_3$ forms has main term $\asymp T^5$.

A third way of viewing this is to look locally. Let $\Pi$ be an automorphic representation of $\mathrm{GL}_3$. At almost all places $v$, the local component $\Pi_v$ is an unramified principal series representation $|\cdot|_v^{t_{v,1}} \boxplus |\cdot|_v^{t_{v,2}} \boxplus |\cdot|_v^{t_{v,3}}$, where the spectral parameters $t_{v,1},t_{v,2},t_{v,3}$ satisfy $t_{v,1} + t_{v,2} + t_{v,3} = 0$ and $|\Re(t_{v,j})| < 1/2$ (the best known bound towards the Ramanujan conjecture in this case is $5/14$ due to Blomer-Brumley). Self-dual representations satisfy the additional (quite stringent!) restriction that $t_{v,j} = 0$ for some $j \in \{1,2,3\}$.