How does the order of random variables matter? Consider two random variables $X_1, X_2$ on sample space $\Omega$. 
Let $P_1, P_2$ are two probability distributions over $X_1, X_2$ for which only order differ. I mean $P_1 (X_1=x_1, X_2=x_2)$ and $P_2 (X_2=x_2, X_1=x_1)$ and all the values are probability values are same.
My doubt in first step is that how to order of random variables matter at all since intersection between sets is commutative?   $P_1 (X_1=x_1, X_2=x_2) = P_2 (X_2=x_2, X_1=x_1) = P(\{\omega\mid X_1(\omega) = x_1\} \cap\{\omega \mid X_2(\omega) = x_2\})$ 
 A: It is worth noting explicitly that you are asking for an explanation of statements made on this Wikipedia page.
The discussion on that page mentions a possibly infinite number of random variables, but even if we consider only the case where there are only two random variables, is not merely about the random variables $X_1$ and $X_2$, it is about the sequence of random variables $(X_1,X_2).$
For $X_1$ and $X_2$ to be exchangeable, the condition is not that
$P_1 (X_1=a, X_2=b) = P_2 (X_2=b, X_1=a).$
The condition is that $P(X_1=a, X_2=b) = P(X_2=a, X_1=b)$:
one probability measure (not two), swapping the values of
the variables.
For example, consider the random variables $X_1$ and $X_2$ with support on the set ${1,2}$ such that
\begin{align}
P(X_1=1,X_2=1) &= \frac13,\\
P(X_1=1,X_2=2) &= \frac13,\\
P(X_1=2,X_2=2) &= \frac13.\\
\end{align}
Then
$$ P((X_1,X_2) = (1,2)) = \frac13 \neq 0 = P((X_2,X_1) = (1,2)) $$
and therefore the variables are not exchangeable.
A: In view of comments posted under the question it seems this may be only a case of misunderstanding something in the linked Wikipedia article.
\begin{align}
\text{These differ from each other: } & \begin{cases} \Pr(X_1=x_1\ \&\ X_2=x_2) \\ \Pr(X_1=x_2\ \&\ X_2=x_1) \end{cases} \\[12pt]
\text{These do not differ from each other: } & \begin{cases} \Pr(X_1=x_1\ \&\ X_2=x_2) \\ \Pr(X_2=x_2\ \&\ X_1=x_1) \end{cases}
\end{align}
A: You are dealing with two different probability measures here. $P_1 (X_1=x_1, X_2=x_2)=P_2 (X_2=x_2, X_1=x_1)$ means $X_1$ and $X_2$ have the same joint distribution under the measures $P_1$ and $P_2$. 
