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.
Wouldn't you rather read: "if $ABC$ is a triangle then the sum of angles $A,B,C$ is 180°" than "$ABC$ is not a triangle or the sum of angles $A,B,C$ is 180°" . Most of theorems in mathematics are of the form "if such and such then you can conclude this and that". Even when there is no antecedent, we usually understand a theorem as meaning "if these axioms hold and if the theorems we have already proved are taken as true, then so on and so forth." You can rewrite this with only "or" and "not", but it will not mean much for most maths students, even mathematicians.