As Brian M. Scott explains, they are logically equivalent.
However, in principle, the expression $$(*) \qquad A \Leftrightarrow B \Leftrightarrow C$$
is ambiguous. It could mean either of the following.
$(A \Leftrightarrow B) \wedge (B \Leftrightarrow C)$
$(A \Leftrightarrow B) \Leftrightarrow C$
These are not equivalent; in particular, (1) means that each of $A,B$ and $C$ have the same truthvalue, whereas (2) means that either precisely $1$ of them is true, or else all $3$ of them are true. Also, you can check for yourself that, perhaps surprisingly, the $\Leftrightarrow$ operation actually associative! That is, the following are equivalent:
- $(A \Leftrightarrow B) \Leftrightarrow C$
- $A \Leftrightarrow (B \Leftrightarrow C)$.
In practice, however, (1) is almost always the intended meaning.