What does $T,t',b \models\phi$ mean? Is it the same as $T,t' \models \phi$ after you have fixed the branch $b$ and $t'$ such that $t' \in b.$ If so wouldn't "for all branches $b'$ through $t$: $T,t,b' \models \phi$" be the same as $T,t \models \phi$.
I am definitely missing something here.
Before I answer, just a remark. Please, when you ask questions like this, specify what your models are.
Modality $\blacksquare$ is a quantifier over paths, or possible timelines, courses of history. Since the definition of a model is not given, I assume that $\mathcal{T}$ is a set of points\states with relation $<$ on them (I also assume that $<$ is a linear order). Now, a history would be a maximal path in $\mathcal{T}$. Note that history is nothing else but sequences of states from $\mathcal{T}$.
So, formula $\blacksquare \varphi$ means that no matter to which history we shift, as long as it includes point $t$, $\varphi$ will hold at $t$ in that history.
It is not the same as $\mathcal{T}, t, b \models \varphi$ as not all paths\histories running through $t$ are the same. We can imagine an example, where $t$ satisfies $p$, and one history carries on to satisfy $p$, while some other history visits some $\lnot p$-states.