A real vector space is any set $V$ endowed with an addition operator $+ : V \times V \to V$ and a scalar multiplication operator $ \cdot : \Bbb R \times V \to V$ such that $+$ is commutative, associative, has an identity $0$ and an inverse $-v$ for each vector $v\in V$, and such that $a(v + w) = av + aw$ for $a\in \Bbb R, v,w \in V$.
It doesn't matter what the elements of the vector space $V$ are. As long as you can define the two operations, it is a vector space, and we can refer to its elements as vectors, and apply the whole theory of vector spaces to it, including the concept of bases. A very VERY common source of vector spaces is spaces of maps. If $S$ is any set, and $W$ is a vector space itself, consider the set V of all maps from $S \to W$. For any two functions $f, g \in V$ and any scalar $a \in \Bbb R$, define $(f+g)(s) = f(s) + g(s)$ and $(af)(x) = af(x)\;\; \forall s \in S$. These make $V$ a vector space. You can choose smaller spaces of maps, provided they are closed under addition and scalar multiplication.
$T_p(\Bbb R^n)$ is just such a vector space. The set $S$ is the set $\mathcal F(\Bbb R^n)$ of smooth functions real-valued functions on $\Bbb R^n$ (or some more local equivalent - it depends on the author which they prefer). The vector space $W = \Bbb R$. And in this case, since $\mathcal F(\Bbb R^n)$ we can, and do, restrict $T_p(\Bbb R^n)$ to linear maps from $\mathcal F(\Bbb R^n) \to \Bbb R$, and further to those that satisfy the Liebnitz condition $v(fg) = f(p)v(g) + g(p)v(f)$. The resulting set of maps $v$ is still closed under addition and scalar multiplication, and so still forms a vector space.
Now consider what this means: an element $v \in T_p(\Bbb R^n)$ is a function that carries functions in $\mathcal F(\Bbb R^n)$ to real numbers. That is, it is an operator on $\mathcal F(\Bbb R^n)$. But because it is also a member of the vector space $T_p(\Bbb R^n)$ it is a vector.
Anything can be a vector, as long as you can define addition and scalar multiplication. That is why vector spaces are so useful. They apply to so many things.
Edit: I did not realize that you had only seen the tangent space defined for $\Bbb R^n$, and where they used the canonical vector space structure of $\Bbb R^n$ to be its own tangent space. In general, manifolds are not vector spaces. When thinking about manifolds, I usually view them as being an undulating surface. But a more concrete example may be helpful to you: the sphere. Consider a point on the sphere and all the various vectors tangent to the sphere at that point. These vectors form a plane that is tangent to the sphere. That plane is the tangent space. (This picture is slightly misleading, as the tangent planes at two different points intersect in $\Bbb R^3$, but we need the actual tangent spaces to be completely disjoint, so we define them a unique abstract objects.) For any curve in the sphere that passes through the point, its derivative is a vector in this plane (all mappings mentioned here are assumed to be smooth). If I have a function $f$ on the sphere and a curve $\phi$ with $\phi(0) = p$ and let $v$ be the tangent vector $\phi'(0)$, then $f \circ \phi$ is from $\Bbb R$ to $\Bbb R$. Further if $\psi$ is a second curve with $\psi(0) = p$ and $\psi'(0) = v$, then you can show that $$\left.\frac{d(f \circ \phi)}{dt}\right |_0 = \left. \frac{d(f \circ \psi)}{dt} \right |_0 $$
In other words, the value of this derivative is dependent only on $v$, not on the curve chosen. Thus we can define the directional derivative $$D_vf(p) = \left.\frac{d(f \circ \phi)}{dt}\right |_0.$$ Note that since $v$ is a tangent vector at $p$, the $p$ in the definition is actually redundant, but I left it in to make it apparent that this is a value is at $p$.
The directional derivative $D_v$ is an operator on the space of smooth functions at $p$. It is linear, and Liebnitzian. In addition to being linear on the functions, it is also linear in $v$. Further, you can show that for any real-valued linear Liebnitzian operator on the space of smooth functions at $p$, there is some vector $v$ such that the operator is $D_v$. That is, the tangent space $T_p(S^2)$ is isomorphic as a vector space to the space of linear Liebnitzian operators at $p$.
A significant problem with the above development is that it depends on how the sphere is imbedded in $\Bbb R^3$. The tangent vectors are vectors in $\Bbb R^3$. But in general what we are interested in with manifolds are their own properties, not the properties of how they sit in another space. So we need a more abstract development of a manifold. But then where do we get the tangent vectors from, if they are not vectors in some containing vector space? While the development of the tangent space as a tangent plane is no longer available, the vector space of linear Liebnitzian operators still is. And since is was isomorphic to the imbedded tangent space, it has exactly the properties we need. That is why in general, the tangent space is defined to be the space of linear Liebnitzian operators at $p$. We just identify each vector $v$ with its directional derivative operator.
$\newcommand{\R}{\mathbb{R}}$
So because we are in $\mathbb{R}^n$ we can all imagine (or at least were told) how we should visualize what a tangent space is at a certain point $p$ in $\mathbb{R}^n$, i.e. some arrows that are tangent. The key point here is that this concept only holds because one can think of the $\mathbb{R}^n$ as vector space (and usually does it). Then a point in $\R^n$ is really nothing else then a vector in $\R^n$. But this is something that will not be true for manifolds. So when you think of $\R^n$ as being a manifold you should not think of any point in $\mathbb{\R}^n$ as vector but really just as point. Now, you can attach a vector space, that is the tangent space, at any point in $\mathbb{R}^n$, and then from there you have a vector space $T_p(\mathbb{\R}^n)$ associated with that point $p$ of the manifold $\R^n$.
Of course now that seems like just a lot of words and really for the $\R^n$ it probably is, but as we get more and more abstract those difference are key in the theory of manifolds.
Now, think of a donut or a sphere in $\R^n$, then again you can attach to any point in that manifold a tangent space. Intuitively you will know how to do it, because that object is imbedded into a euclidian space. But later on in your studies of manifolds this will not be the case! (At least not in any practical way).
So, we need to find a concept of tangent vectors that only rely on the local properties of a point of a manifold. And this is were the (partial) derivative notation comes into play. Because these are concepts that can be also defined for manifolds in general.
To the question at hand: As it was pointed out, isomorphic means it is the same thing! Both are $n$ dimensional linear vector space which are isomorphic to each other and now the only question is which kind of "names" you want to give these vectors. But that is all that is different: the symbols.
Edit: Maybe to make it more precise: Take the basis $e_1,\dots,e_n$ of $T_p(\R^n)$ and let $\varphi : T_p(\R^n) \to D_p(\R^n)$ be the isomorphism. Then $\varphi(e_i) = \partial/\partial_{x^i}|_p$ for all $i\in\{1,\dots,n\}$. Now, you do the calculations for this basis in $D_p(\R^n)$. So, whenever you now have a element $v \in D_p(\R^n)$ and you had enough of this derivative notation you simply do $\varphi^{-1}(v)$ or even more concrete: you have $v = \sum_{i=1}^n v^i \partial/\partial_{x^i}|_p$, so $\varphi^{-1}(v) = \sum_{i=1}^n v^i e_i$.
Best Answer
The tangent space at a point $p$ is not defined via $D_v$ as v ranges over all directions. It is abstractly defined as the space of linear maps $H: C^{\infty}(U) \mapsto \mathbb{R}$ which obey a product rule, i.e. $H(fg)=f(p)H(g)+g(p)H(f)$. Hence, it is not apparent that every such member can be identified with a directional derivative