Question: I came across the equivalences of conditionals and biconditionals. In the conditionals case, it's $P \implies Q \equiv ( \lnot P) \lor Q$, and the biconditional is $P \iff Q \equiv ( \lnot P \land \lnot Q ) \lor (P \land Q )$.

My question is, if both conditionals and biconditionals can be boiled down to negations, conjunctions and disjunctions, then why are conditionals and biconditionals used in logic?

I imagined at first that it's sort of like programming, where it would simplify control flow, but then I learned there aren't any more "complex" connectives like them, so why are those two special enough to use? Why aren't there more symbolizations of complex propositional forms?

Answer: 
Please note that all truth-functional expressions can be boiled down to NAND's. So given that, you might as well ask: why are we even using conjunctions, disjunctions, and negations?

Also note that I should be able to program all computers by directly writing down long strings of $0$'s and $1$'s. So why do we have languages like C++, Python, or JavaScript?

Why, indeed, talk about planets when ultimately they are nothing but big collections of atoms? Hell, once I know all fundamental forces of physics, why even do chemistry, biology, cognitive science, sociology, etc?

I hope you start to see the answer: it is super useful to have 'higher-level' or 'macro-level' perspectives. Indeed, given the inherent cognitive limitations of the human mind, we really *need* those. Think about it: why do we do statistics, when all it amounts to is lossy data compression? Ultimately it is because we are of limited intelligence, because if we were God, we would have no need for statistics. In general, an infinitely smart being can look at the raw 'low-level' descriptions, and 'see' everything there is to divine. As humans, however, our explanations, predictions, and general analyses, abilities, and understanding is greatly enriched by high-level concepts and perspectives.
Although it is true that the conditional and biconditional can be boiled down to simpler functions in classical two-valued logic, this may be misleading when attempts are made to extend logic beyond the classical limits.

The biconditional expresses that two propositions are equally true or false, or have the same truth value. In the theory of mathematical structure, this has the properties of an equivalence relation. (Reflexive, Symmetric, Transitive)

The conditional expresses that the conclusion is not less true than the hypothesis. In the theory of mathematical structure, this has the properties of an ordering relation. (Reflexive, Antisymmetric, Transitive)

These properties make extended chains of reasoning possible.

In variants of logic, for instance multivalued logics, modal logic, or intuitionism, the neat correspondence between the conditional and the biconditional and the other logical operations (negation, conjunction and disjunction) breaks down. The conditional and biconditional have to be otherwise defined or their properties somehow asserted independently in order for them to be useful.