I do not have a background in formal logic, but from the point of view of mathematics outside formal logic (which is the point of view from which Tao's book is written, for example), "Prove $P(x)$ and $Q(x)$ are equivalent" means the same thing as "Prove $P(x)$ and $Q(x)$ imply each other." There is no difference between the equivalence and the biconditional.
However, in all cases, you have to show more than that the biconditional holds when the statements hold. That's vacuous. If $P(x)$ and $Q(x)$ both hold, then $P(x)\leftrightarrow Q(x)$ holds too, just by truth-table reasoning. What you have to prove is that $P(x)\leftrightarrow Q(x)$ in all situations. Another way to say this is that you need to prove that whenever $P(x)$ holds, $Q(x)$ also holds, and whenever $Q(x)$ holds, $P(x)$ also holds.
In practice, this is not usually done by considering truth tables, but instead just by reasoning directly with the statements. For example:
Suppose $A\subseteq B$. Then, unioning both sides with $B$ and observing that this operation preserves the relation of containment, we can conclude that $A\cup B\subseteq B\cup B$. But $B\cup B=B$, and we can conclude $A\cup B\subseteq B$. Meanwhile, Also $A\cup B\supseteq B$ since $A\cup B$ is a union of $B$ and something else. Since $A\cup B\subseteq B$ and $A\cup B\supseteq B$, we can conclude $A\cup B=B$. We have thus proven that $A\subseteq B\Rightarrow A\cup B = B$.
You do a similar thing to show that $A\cup B = B \Rightarrow A\subseteq B$, etc.
Addendum: Per the discussion below in comments, I think the heart of the matter is that while it might at first appear that proving the two implications "P implies Q" and "Q implies P" is strictly weaker than proving equivalence, it is actually no weaker!
"P implies Q" is the same thing as "It is impossible for $P(x)$ to hold without $Q(x)$ also holding." Meanwhile, "Q implies P" is the same thing as "It is impossible for $Q(x)$ to hold without $P(x)$ holding." Thus, if you prove both implications, you have established that it is impossible for $P$ and $Q$ to have different truth values in any model. In other words, they have the same truth value in every model. This reasoning is not contingent on assuming they have the same truth value: the claim that they always have the same truth value follows as a consequence of the pair of statements "P implies Q" and "Q implies P".