In 1905, Lebesgue gave a "proof" of the following theorem:
If $f:\mathbb{R}^2\to\mathbb{R}$ is a Baire function such that for every $x$, there is a unique $y$ such that $f(x,y)=0$, then the thus implicitly defined function is Baire.
He made use of the "trivial fact" that the projection of a Borel set is a Borel set. This turns out to be wrong, but the result is still true. Souslin spotted the mistake, and named continuous images of Borel sets analytic sets. So a mistake of Lebesgue led to the rich theory of analytic sets. Lebesgue
seemingly enjoyed this fact and mentioned it in the foreword to a book, "Leçons sur les Ensembles Analytiques et leurs Applications", of Souslin's teacher Lusin (as referenced in an AMS review of the book).
This isn't an exact analogue to P != NP, in which two large classes exist and it is undecided whether they are equal or not; instead, two large "universes" exist, of which only one is the truth, with one of them strongly believed to not exist, but for which all attempts to disprove this parallel universe have been defeated by an invisible fence. (Perhaps a complexity theory analogue would be a scenario in which we knew that of Impagliazzo's five worlds, only Algorithmica or Cryptomania were possible, but we could not determine which, with both worlds showing an equal propensity to "want to exist".)
Anyway, the situation is in analytic number theory, where there are two worlds (which, very roughly, would correspond to "Algorithmica" and "Cryptomania" in Impagliazzo's list):
Siegel zero: The primes conspire (i.e. show extremely anomalous correlation) with some multiplicative function, such as a Dirichlet character $\chi$; roughly speaking, this means that there is some modulus q such that there is a huge bias amongst the primes to be quadratic nonresidues mod q rather than quadratic residues. (Dirichlet's theorem tells us that the bias will die down eventually - for primes exponentially larger than q - but this is not useful in many applications). The most common way to describe this scenario is through a "Siegel zero" - a zero of an L-function that is really, really far away from the critical line (and really close to 1). Weirdly, such a conspiracy actually makes many number theory problems about the primes easier than harder, because one gets to "pretend" that the Mobius function is essentially a character. For instance, there is a cute result of Heath-Brown that if there are an infinite family of Siegel zeroes, then the twin prime conjecture is true. (Basically, the principle is that at most one conspiracy in number theory can be in force for any given universe; a Siegel zero conspiracy sucks up all the "conspiracy oxygen" for a twin prime conspiracy to also hold.) It does lead to some other weird behaviour though; for instance, the existence of a Siegel zero forces many of the zeroes of the Riemann zeta function to lie on the critical line and be almost in arithmetic progression.
Standard model: this is the universe which is believed to exist, in which the primes do not exhibit any special correlation with any other standard multiplicative function. In this world, GRH is believed to be true (and the zeroes should be distributed according to GUE, rather than in arithmetic progressions (this latter hypothesis has occasionally been called the "Alternative hypothesis")).
[This is an oversimplification; much as how complexity theorists have not ruled out the intermediate worlds between Algorithmica and Cryptomania, we don't have as strong of a separation between these two number-theoretic worlds as we would like. For instance there could conceivably be intermediate worlds where there are no Siegel zeroes, but GRH or GUE still fails (somewhat analogous to Impagliazzo's "Pessiland"). So in practice we have to weaken one or the other of these worlds, for instance by replacing GRH with a much weaker zero-free region. I'm glossing over these technical details though for this discussion. My feeling is that we have some chance with current technology of eliminating some more of these intermediate worlds, but we're quite stuck on eliminating either of the extreme worlds.]
In many different problems in analytic number theory, one has to split into two cases depending on which of the above universes is actually the one we live in, and use different arguments for each (which is the major source of the notorious ineffectivity phenomenon in analytic number theory, that many of out estimates involve completely ineffective constants, as they could depend on the conductor of a Siegel zero, for which we have no upper bound); finding a way around this dichotomy in even just one of these problems would be a huge breakthrough and would likely lead soon to the elimination of one of these universes from consideration. But there is an invisible fence that seems to block us from doing so; both universes exhibit a surprising amount of "self-consistency", suggesting that one could modify our mathematical universe very slightly one way or the other to "force" one of the two scenarios to be in effect (a bit like how one can force P to equal or not equal NP by relativising). The "parity barrier" in sieve theory is one big (and relatively visible) section of this fence, but this fence seems to be much longer and more substantial than just this parity barrier.
I discuss these things a bit more in this blog post. See also this survey of Conrey that was also linked to above.
Best Answer
For a quite extensive overview with many examples, you might want to check out The origins and development of mathematical notation. I also enjoyed reading Stephen Wolfram's take on Mathematical Notation – past and future, with a great variety of illustrations.
The OP asks specifically for the evolution of one formula, "to see the big picture". Here is one example, taken from Math through the ages: