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: 
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.
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.