$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\supp}{\operatorname{supp}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$
Weak solutions (aka generalized solutions) occupy a place between ordinary regular solutions and solutions which are distributions, usually when the existence of the former is not proven (or proven that does not have place) and the latter does not make sense.
Weak solutions must satisfy certain integral identities which for regular solutions are equivalent to original equations. These identities invoke an arbitrary test function (that means this identity must hold for all test functions; this space of test functions plays important role.
Often this weak solution is obtained as a weak limit of regular solutions $u_k$: $u_k \overset{w}{\rightarrow} u$ that is $\langle u_k,\phi\rangle \to \langle u, \phi\rangle $ for all test functions (this space of test functions may differ from the previous one). Sometimes this weak solution is obtained as strong limit of regular solutions $u_k$: $u_k \overset{s}{\rightarrow} u$ where $u$ belongs to some Banach space and convergense is in this space (that is $|u_k -u|_B\to 0$). All this clearly belongs to Real Analysis and PDEs were one of the main reasons to develop it.
The notion of the weak solution is deliberately left fuzzy since it is context dependent: in differnt cases we use different definitions. We start from equations without boundary conditions so most likely there will be no uniqueness.
Example 1.
Example 2. Consider equation (\ref{eq-11.4.3}) again but now we add condition \begin{equation} \bigl[-\sum_{jk} a_{jk}u_{x_j}\nu_k-\frac{1}{2}\sum_{j}b_{j}u\nu_j\bigr]\bigr|_\Gamma=h \label{eq-11.4.5} \end{equation} where $\nu_k$ are components of the unit inner normal to $\Gamma$. Then in the smooth case (\ref{eq-11.4.3}) and (\ref{eq-11.4.5}) together are equivalent to \begin{equation} Q[u,\varphi]= \iint_\Omega f\varphi\,dx +\int_\Gamma h\varphi\,dS \label{eq-11.4.6} \end{equation} which should hold for all $\varphi\in C^1(\mathbb{R}^d)$ (now $\varphi$ is not necessarily $0$ near $\Gamma$!) and we call $u\in C^1(\Omega)$ (or even worse) weak solution to problem (\ref{eq-11.4.3}), (\ref{eq-11.4.5}) in this case. Here $Q[u,\varphi]$ is the left-hand expression in (\ref{eq-11.4.4}).
These examples could be extended to quasilinear or even nonlinear equation. See Burgers equation in Section 12.1.
Example 3. Special case of Example 2: \begin{gather} Q[u,v]:= -\iint u \partial_j v \,d^nx= \iint u^{(j)} v\, dx^n ; \label{eq-11.4.7} \end{gather} then $u^{(j)}$ is a generalized derivative of $u$ and denoted as $\partial_j u$.
More general
\begin{gather}
Q[u,v]:= \iint u (-\partial)^\alpha v \,d^nx= \iint u^{(\alpha)} v\, dx^n
\label{eq-11.4.8}
\end{gather}
with multiindex $\alpha=(\alpha_1,\ldots,\alpha_n) \in \mathbb{Z}^{+n}$ and
$\partial^\alpha:=\partial_1^{\alpha_1}\cdots \partial_n^{\alpha_n}$, $\partial_j:=\partial_{x_j}$.
then $u^{(\alpha)}$ is a generalized higher-order derivative of $u$ and denoted as $\partial^\alpha u$
Are they different? If yes, which are more general and more inportant.
Yes, they are different despite significant overlapping and both are very important. While solutions which are distributions cover completely linear PDEs with smooth coefficients they do not work in non-linear PDEs since non-linerar function of distribution is not defined (f.e. $\delta(x)$ multiplied by $\delta(x)$ os $\sin (\delta(x))$ are not defined. Sure there are calculi where those are defined but they are completely marginal efforts. Moreover, product of $\delta(x)$ and $\theta(x)$ (step function at $0$ are not defined either).
Also distributional solutions can be ill-suited for boundary problems if boundary is not smooth. Still if $u$ is a bounded measurable function (see Real Analysis) we can calculate distributional derivatives of $u(x)^2$ etc, so such functions can be soluions in the distributional sense.
Example 4. Stationary points of functionals of Chapter 10 appear as weak solutions of boundary value problems. Sometimes convexity properties of the functional prove that those are strong solutions. But gigorous proof that they are regular solutions are highly non-trivial.
Example 5. Consider \begin{align} &u_{tt} - u_{xx} =f, &&t>0,\label{eq-11.4.9}\\ &u|_{t=0}= g, &&u_t|_{t=0}=h,\label{eq-11.4.10}\\ &u|_x|_{x=0}= \phi(t).\label{eq-11.4.11} \end{align} If $u,v$ are smooth functions and $v=0$ as $|x|+|t|>A$ then \begin{multline*} \iint_{x>0,t>0} (u_{tt}-u_{xx})v \, dxdt = \iint_{x>0,t>0} u(v_{tt}-v_{xx}) \, dxdt\\ + \int_{x>0,t=0} (-u_tv+uv_t)\,dx + \int_{x=0, t>0} (u_x v -uv_x )\,dt. \end{multline*} Then for $f=g=h=\phi=0$ we should have \begin{gather*} 0 = \iint_{x>0,t>0} u(v_{tt}-v_{xx})v\, dxdy- \int_{x=0, t>0} uv_x \,dt. \end{gather*} Let us plug $u=1$ as $t>x>0$, $u=0$ as $x>t>0$ which is a candidate for solution in Section 2.5.
We get \begin{gather*} 0 = \iint_{t>x>0} (v_{tt}-v_{xx}) \, dxdy- \int_{x=0, t>0} v_x \,dt. \end{gather*} But the first integral is equal \begin{gather*} \int_{x=t} (v_x+v_t) \,dx = -v(0,0) \end{gather*} a and we do not get $0$ for all indicated $v$ and $u$ is not a weak solution of problem (\ref{eq-11.4.9})-(\ref{eq-11.4.11}) with $f=g=h=\phi=0$.
Meanwhile, similar calculations show that the same $u$ is a weak solution for the problem (\ref{eq-11.4.9})-(\ref{eq-11.4.10}) wth the condition \begin{gather} u|_{x=0}=\psi(t) \label{eq-11.4.12} \end{gather} with $\psi(t)=1$. Thus if $f,g,h,\phi$ are continuous functions then weak solution of (\ref{eq-11.4.9})-(\ref{eq-11.4.11} is continuous in $0$. For problem (\ref{eq-11.4.9})-(\ref{eq-11.4.10}, (\ref{eq-11.4.12}) it is continuous if and only if compatibility codition \begin{gather} g(0)=\psi(0) \label{eq-11.4.13} \end{gather} is fulfilled.