$P(X_2=1 | X_1=3)$ is the probability that $X_2=1$ given that $X_1=3$. This is called a conditional probability. You assume that $X_1=3$, and from there, compute the probability that $X_2=1$.
$P(X_2=1 , X_1=3)$ is the probability that $X_2=1$ and $X_1=3$. Here, you are computing the probability that both of the events occur simultaneously.
They are quite different:
Example:
Toss a coin twice in succession. The probability that the first flip is a head and the second flip is a head is $1/4$. But, the probability that the second flip is a head, given that the first flip is a head, is 1/2.
But they are related: $ \tag{1}P(X_2=1 | X_1=3)={P(X_2=1 , X_1=3)\over P(X_1=3)}, \text{ if } P(X_1=3)\ne0. $
The notation $P(X_2 = 1 \wedge X_1=3)$ is just a different way of writing $P(X_2 = 1 , X_1=3)$.
Edit:
A word about formula (1) (warning: I'm hand-waving here). Given that the event $B$ happens, your sample space is "reduced" to those outcomes and only those outcomes that are in $B$. Then, within this reduced sample space, the probability of the event $A$ is a certain proportion of the reduced sample space. What is the proportion? Well, it's the the one given by formula (1).
This may be seen more easily by considering a Venn diagram, where probabilities are interpreted as areas. $P(A|B)$ is the ratio of the area of $A\cap B$ with the area of $B$.