What's the difference between biconditional iff and logical equivalence?

In short, $P \leftrightarrow Q$ is statement that could be either true or false. $P \equiv Q$ means that $P \leftrightarrow Q$ is always a true biconditional (so, $P$ and $Q$ have the same truth value no matter what).

So, one could say that $\neg (P \vee Q) \equiv \neg P \wedge \neg Q$ (DeMorgan's) but you typically wouldn't write $\neg (P \vee Q) \leftrightarrow (\neg P \wedge \neg Q)$.

The arrow $\Rightarrow$ usually is slang for "implies" but different people use it differently. The arrow $\Leftrightarrow$ is usually treated the same way as $\leftrightarrow$.

In case you can use a somewhat philosophical explanation: $\leftrightarrow$ is a logical operator within statements, while $\equiv$ serves to state an equivalence between statements and thus may be thought of as meta-logical.

As Randall explained, $P \leftrightarrow Q$ is a statement $-$ one statement and a logical statement. The $\leftrightarrow$ will cause it to be true under certain truth value distributions for $P$ and $Q$. The same applies to $\neg(P \vee Q) \leftrightarrow (\neg P \wedge \neg Q)$.

For $\neg(P \vee Q) \equiv (\neg P \wedge \neg Q)$ however, you compare two truth tables, the one of $\neg(P \vee Q)$ and $(\neg P \wedge \neg Q)$, two distinct statements. If and only if both are true exclusively for the same truth value distributions, the equivalence applies and so the meta-logical statement $\neg(P \vee Q) \equiv (\neg P \wedge \neg Q)$ is true.

Consider this analogy. You wouldn't say the following:

Prove that 3 + 5.

You might instead say this:

Prove that 3 = 5.

Similarly, following doesn't make sense:

Prove that $p \leftrightarrow q$.

Instead one of these would be correct:

Prove that $p \leftrightarrow q$ is always true.
or
Prove that $p \equiv q$.