What can we infer from Hopf's lemma and $\frac{\partial u}{\partial\nu}(x_0)>0$? Let $\Omega\subseteq\mathbb R^d$ be open, $u\in C^2(\Omega)$ be subharmonic and $$M:=\sup_\Omega u.$$ We then know that if $$C:=\{u=M\}\ne\emptyset$$ and $\Omega$ is connected, then $u\equiv M$. This is known as the "maximum principle".
Now assume $\Omega$ is an open ball and $u\in C^1(\overline\Omega)\cap C^2(\Omega)$ is subharmonic. Moreover, let $M\in\mathbb R$ with $$u<M\;\;\;\text{in }\Omega\tag1.$$ By Hopf's lemma, if $u(x_0)=M$ for some $x_0\in\partial\Omega$, then $$\frac{\partial u}{\partial\nu}(x_0)>0\tag2.$$

The important aspect is the strict inequality in $(2)$, since by $(1)$ we already know that $x_0\in\partial\Omega$ is a maximum of $u$ and hence $T_{x_0}(u)v\ge0$ for every outward pointing $v\in T_{x_0}\overline\Omega$.
However, what is the interpretation of $(2)$? Yes, it states that the derivative of $u$ in the direction of the normal field $\nu$ at $x_0$ is positive, but what can we infer from that?

 A: The maximum principle you refer to is generally referred to as the strong maximum principle, and according to Gilbarg & Trudinger this is also a result of Hopf - I wasn't able to find the original paper online, but I imagine Hopf's lemma was historically used as an intermediate step in proving said result (and that's how it is presented in most texts).
In light of this, I think of Hopf's lemma as the technical essence of the strong maximum principle. The main idea behind the maximum principle is that a subharmonic function cannot have a strict maximum in the sense that the Hessian $\nabla^2u$ is positive definite by considering signs. However this still allows maxima where $\nabla^2u$ is non-negative definite, and the lack of strictness is the main difficulty in establishing the stronger statment (one can use a perturbation argument to get a weak statement however).
Hopf's lemma is the key step to overcome this difficulty, providing a strictly positive quantity that allows one to deduce a contradiction via similar means. Here we note that if $u$ is harmonic which is non-constant near an interior maximum point $x_0,$ then $\nabla u(x_0)$ must necessarily be non-constant, contradicting the fact that it is a maximum point. Again, the strictness is key to obtain a contradiction.

The fact that it is the "technical essence" also means it's useful in applying the lemma in situations where the strong maximum principle doesn't quite apply, but almost does. One simple example is the uniqueness of the Neumann problem; if $u \in C^2(\Omega) \cap C^1(\overline{\Omega})$ is harmonic in $\Omega$ which is a sufficiently regular domain with $\frac{\partial u}{\partial \nu} = 0$ on $\partial\Omega,$ then it is easy to infer that $u$ is constant using Hopf's lemma.
