This further answer is meant as a feedback to dividebyzero's comment to my previous answer.
As dividebyzero says, it is certainly true that CNF and DNF are two sides of the same coin.
When you have to find a satisfying assignment, DNF is explicit as it manifestedly shows you its satisfying assignments (DNF Satisfiability belongs to P), whereas CNF is implicit as it wraps and winds to hide its satisfying assignments from your eyes (CNF Satisfiability is NP−complete). We do not know any procedure which is able to unwrap and unwind any CNF formula into some equisatisfiable DNF formula which is not exponentially sized. This was the point of my previous answer (whose example was meant to show the exponential blow-up, although admittedly such example was not the best possible choice).
Conversely, when you have to find a falsifying assignment, CNF is explicit as it manifestedly shows you its falsifying assignments (CNF Falsifiability belongs to P), whereas DNF is implicit as it wraps and winds to hide its falsifying assignments from your eyes (DNF Falsifiability is NP−complete). We do not know any procedure which is able to unwrap and unwind any DNF formula into some equifalsifiable CNF formula which is not exponentially sized.
At one extremity we have Contradictions, i.e. unsatisfiable formulas. At the opposite extremity we have Tautologies, i.e. unfalsifiable formulas. In the middle, we have formulas which are both satisfiable and falsifiable.
In any CNF formula with n variables, every clause of length k manifestedly encodes 2n−k falsifying assignments.
In any DNF formula with n variables, every term of length k manifestedly encodes 2n−k satisfying assignments.
A CNF formula without clauses is a Tautology, because it does not have any falsifying assignment. A CNF formula containing the empty clause (which subsumes every other clause) is a Contradiction, because the empty clause (which has k=0) indicates that all the 2n assignments are falsifying. Any other CNF formula is either a Contradiction or one of those formulas in the middle (and it is NP−complete to distinguish between these 2 cases).
A DNF formula without terms is a Contradiction, because it does not have any satisfying assignment. A DNF formula containing the empty term (which subsumes every other term) is a Tautology, because the empty term (which has k=0) indicates that all the 2n assignments are satisfying. Any other DNF formula is either a Tautology or one of those formulas in the middle (and it is NP−complete to distinguish between these 2 cases).
With a CNF formula, distinguishing between the 2 cases above means being able to tell whether all the falsifying assignments collectively brought by the presence of clauses overlap in such a way to cover all the 2n assignments (in which case the formula is a Contradiction, otherwise it is satisfiable).
With a DNF formula, distinguishing between the 2 cases above means being able to tell whether all the satisfying assignments collectively brought by the presence of terms overlap in such a way to cover all the 2n assignments (in which case the formula is a Tautology, otherwise it is falsifiable).
Under this light it becomes more clear why CNF Satisfiability and DNF Falsifiability are equivalent in terms of computational hardness. Because they actually are the very same problem, as the underlying task is exactly the same: to tell whether the union of several sets equals the space of all possibilities. Such task leads us to the wider realm of counting, which is in my humble opinion one of those avenues to be fervently explored in order to hope to make some non-negligible progress on these problems (I doubt that further research on resolution-based solvers may eventually bring groundbreaking theoretical advancements, while it certainly continues to bring surprising practical advancements).
The difficulty of such task is that those sets overlap wildly, in an inclusion - exclusion fashion.
The presence of such overlapping is precisely where the hardness of counting resides. Moreover, the fact that we let those sets overlap is the very reason that allows us to have compact formulas whose solution space is nevertheless exponentially large.