There are various ways to explain this, but probably the best way to start is to try to think of "continuous at a point" or "limit at a point" as being its own independent concept, rather than something defined in terms of right-continuity and left-continuity. In the case of the real numbers, there are a lot of ways to define this, but here are two good ones:
Let $D\subseteq \Bbb R$ and let $f\colon D\to \Bbb R$. Then $f$ is continuous at $c\in D$ iff:
For every $\epsilon > 0$ there is a $\delta >0$ such that for all $x\in D$ such that $|x-c|<\delta$, $|f(x)-f(c)| < \epsilon$.
Whenever $(x_i)$ is a sequence in $D$ that converges to $c$, $(f(x_i))$ converges to $f(c)$.
Edit:
Thus the notion of continuity at an endpoint is perfectly sensible. Also, it's quite reasonable to consider a derivative at an endpoint. I think you may be getting a little confused because there are so many theorems out there that apply when a function is continuous on an interval and differentiable in its interior—it's not that the function can't be differentiable there, but that the theorem doesn't need it to be.
It is not that "closed intervals are used for continuity and open intervals for differentiability" (more on this one later). It is that, for Rolle's Theorem (and the Mean Value Theorem), we need those hypotheses.
In the proof, we use that a continuous function on $[a,b]$ attains a maximum. And we only need differentiability inside, so we do not need to make further assumptions on the boundary about differentiability (again, more on this later). And it is a nice exercise to see that if you relax any hypothesis on Rolle's Theorem you do not have a true general statement anymore.
Now, continuity can be talked about in far more general settings. More particularly, we can talk about continuity on any subset of the real numbers in a rather canonical fashion (no need to be intervals, closed or open or whatever).
Differentiability is a little trickier. It is common to define differentiability only on open sets when we are in Euclidean space (not only open intervals, but open sets in general). This is partly due to the fact that being able to differentiate from every direction is a must in some theorems and some basic facts which we would like to have. However, there are cases for which talking about differentiability, in some sense, on "not-open" sets is useful and/or a must. This is true for example when talking about functions on the closed half-space (which enhances its discussion on manifolds with boundaries), or when talking about closed submanifolds of some manifold.
In your particular setting, we can define differentiability on $[a,b]$ on many ways. Firstly, we can simply extend to the fact that the limit which defines the derivative exists on the boundaries (however, it will be only a one-sided limit). Or we can extend by saying that $f$ is differentiable on $[a,b]$ if there exists a differentiable function $g$ on an open set containing $[a,b]$ such that $g|_{[a,b]}=f$. Instead of discussing this further, I'll just say that differentiability is more subtle than continuity with respect to its domains.
Best Answer
The problem is one of consistent definitions. Intuitively we can make sense of differentiable on a closed interval: but it requires a slightly more careful phrasing of the definition of "differentiable at a point". I don't know which book you are using, but I am betting that it contains (some version of the) following (naive) definition:
To make sense of the limit, often times the textbook will explicitly require that $f$ be defined on an open interval containing $x$. And if the definition of differentiability at a point requires $f$ to be defined on an open interval of the point, the definition of differentiability on a set can only be stated for sets for which every point is contained in an open interval. To illustrate, consider a function $f$ defined only on $[0,1]$. Now you try to determine whether $f$ is differentiable at $0$ by naively applying the above definition. But since $f(y)$ is undefined if $y<0$, the limit
$$ \lim_{y\to 0^-} \frac{f(y) - f(0)}{y} $$
is undefined, and hence the derivative cannot exist at $0$ using one particular reading of the above definition.
For this purposes some people use the notion of semi-derivatives or one-sided derivatives when dealing with boundary points. Other people just make the convention that when speaking of closed intervals, on the boundary the derivative is necessarily defined using a one-sided limit.
Your textbook is not just being pedantic, however. If one wishes to study multivariable calculus, the definition of differentiability which requires taking limits in all directions is much more robust, compared to one-sided limits: the main problem being that in one dimension, given a boundary point, there is clearly a "left" and a "right", and each occupies "half" the available directions. This is no longer the case for domains in higher dimensions. Consider the domain
$$ \Omega = \{ y \leq \sqrt{|x|} \} \subsetneq \mathbb{R}^2$$
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A particular boundary point of $\Omega$ is the origin. However, from the origin, almost all directions point to inside $\Omega$ (the only one that doesn't is the one that points straight up, in the positive $y$ direction). So the total derivative cannot be defined at the origin if a function $f$ is only defined on $\Omega$. But if you try to loosen the definitions and allow to consider only those "defined" directional derivatives, they may not patch together nicely at all. (A canonical example is the function $$f(x,y) = \begin{cases} 0 & y \leq 0 \\ \text{sgn}(x) y^{3/2} & y > 0\end{cases}$$ where $\text{sgn}$ return $+1$ if $x > 0$, $-1$ if $x < 0$, and $0$ if $x = 0$. Its graph looks like what happens when you tear a piece of paper.)
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But note that this is mainly a failure of the original naive definition of differentiability (which, however, may be pedagogically more convenient). A much more general notion of differentiability can be defined:
This definition is a mouthful (and rather hard to teach in an introductory calculus course), but it has several advantages:
For this definition, you can easily add
Note how this looks very much like the statement you quoted in your question. In the definition of pointwise differentiability we replaced the condition "$x$ is contained in an open neighborhood" by "$x$ is a limit point". And in the definition of differentiability on a set we just replaced the condition "every point has an open neighborhood" by "every point is a limit point". (This is what I meant by consistency: however you define pointwise differentiability necessarily effect how you define set differentiability.)
If you go on to study differential geometry, this issue manifests behind the definitions for "manifolds", "manifolds with boundaries", and "manifolds with corners".