What is a differetial structure, exactly? A structure in general is a set and some operations on that set or ordersrelations of some kind.
In algebra and topology this is rather clear, but in differential geometry one often consider "differential structure".
I understand that an atlas is related to this matter and my impression is that an atlas induces a differential structure. But what is this structure?
In topology we can relate the structure to continuity i.e any topological space space that has the same topological structure also have the same continuous functions(even if it is not defined on the exact same sets). Hence one would think that differential structure and differentiable functions have the same dynamics. 
Something tells me this is related to the tangent spaces and how they look as linear spaces since this is what the atlas induces at each point via the partials of the charts.
Does anyone have  good answer for what the differential structure consists of or how to think about it? Two different manifolds with different atlases should be able to have the same "differential structure" as far as I understand.
 A: Definitions:
Definition: Let $n$ be a non-negative integer number and let $M$ be a topological space. We say that $M$ is a topological $n$-manifold if it satisfies the following axioms.
1) $M$ is Hausdorff.
2) $M$ is second-countable, that is, there exists an at most countable basis for the topology.
3) $M$ is locally euclidean, that is, for every point $p$ in $M$ and every open neighborhood $U$ of $p$ there exists an homeomorphism $\varphi : V \to W$, where $V$ is an open neightborhood of $p$ contained in $U$ and $W$ is an open set of $\mathbb{R}^n$. 
In such case, a pair $(\varphi,V)$, where $\varphi:V\to W$ is an homeomorphism from an open set $V$ of $M$ to an open set $W$ of $\mathbb{R}^n$, is called a chart of $M$.
Remark: For $n=0$, we agree $\mathbb{R}^0=\{0\}$ is a singleton endowed with the unique possible topology so a topological $0$-manifold is the same as an at most countable discrete space.
Definition: A smooth atlas, or more briefly an atlas, on a topological $n$-manifold $M$ is a set $\mathcal{A} = \{(\varphi_j,V_j)\mid j\in J\}$ of charts of $M$ that satisfy the following conditions.
1) The set $\{V_j\mid j\in J\}$ is an open cover of $M$, that is, $\cup_{j\in J}V_j=M$.
2) For every $i$ and $j$ in $J$ such that $V_i\cap V_j\neq\emptyset$ we have that $\varphi_i\circ \varphi_j^{-1} : \varphi_j(V_i\cap V_j)\to \varphi_i(V_i\cap V_j)$ is smooth, that is, infinitely differentiable.
Remark: Given a topological $n$-manifold $M$ such that there exists an smooth atlas on it, we can consider the poset (partial ordered set) of the set of atlas on $M$ ordered by inclusion.
Definition: A smooth $n$-manifold is a pair $(M,\mathcal{A})$, where $M$ is a topological $n$-manifold and $\mathcal{A}$ is a maximal smooth atlas on $M$.
Lemma: Let $M$ be a topological $n$-manifold. For every smooth atlas $\mathcal{A}$ on $M$ there exists a unique maximal atlas $\mathcal{A}^{\text{max}}$ on $M$ containing $\mathcal{A}$. Given two smooth atlas $\mathcal{A}_1$ and $\mathcal{A}_2$ on $M$ we have that $\mathcal{A}_1^{\text{max}}=\mathcal{A}^{\text{max}}_2$ if and only if $\mathcal{A}_1\cup\mathcal{A}_2$ is an atlas.
For a proof see Proposition 1.17 in Lee.
Definition: A differential structure on a topological $n$-manifold $M$ is a maximal atlas $\mathcal{A}$ on $M$. We say that two atlas $\mathcal{A}_1$ and $\mathcal{A}_2$ on $M$ determine the same differential structure if $\mathcal{A}_1^{\text{max}}=\mathcal{A}^{\text{max}}_2$.
Comments:
1) If you like to adopt an algebraic point of view, we could compare the properties of the poset of atlas on a topological manifold with the poset of topologies on a set and they are very different.
2) The atlas on manifolds say what functions between manifolds are smooth in the same sense that topologies say what functions between topological spaces are continuous. Smooth manifolds and smooth functions form a category and the same is true for topological spaces and continuous functions. In a category we have a notion of isomorphism which says that two objects are equivalent and, as @LeeMosher pointed out in the comments, in many cases it admits an interpretation in terms of preserving the structure. In the first case, an isomorphism is a diffeomorphism and, in the second case, it is an homeomorphism.
