Although the words have superficially similar pronunciations, performativity and performance are two extremely different notions. In her book Gender Trouble (1990) and its sequel Bodies That Matter (1993), Judith Butler put forth the thesis that gender identity is performative. Over the last decade performance-based theories of gender and identity have become popular, even mainstream, despite a number of deep-seated and readily-apparent flaws. Unfortunately, these latter performance-based theories are often portrayed as successors of Butlerean performativity. They're not.
To understand performativity one should go back to Austin's original definition of performative speech acts. Whenever we speak, we speak for a reason. Austin was interested in explaining these reasons— in particular, explaining the contrast between what we say and why we say it. When we ask "could you pass the salt?" we are not literally interested in whether the addressee is capable of moving the salt shaker, we're making a request. When we ask "how do you do?" or "what's up?" we do not actually want an answer, we are merely greeting someone. It is within this context of discussing the why behind what we say that Austin became interested in performative speech acts: speech acts which through their very utterance do what it is they say, or speech acts which are what it is they mean. When the right person in the right context utters "I now pronounce you married", that vocalization is in fact the pronouncement itself. To state that you pronounce something, is itself to make the proclamation. In just the same way, when under the right circumstances someone says they promise such-and-so, they just did.
There are a number of interesting details about what it means to be a performative speech act. For instance, just uttering the words is not enough: if a random stranger comes up to you and pronounces you married, that does not actually mean you're married. For the performative speech act to have any force it must be uttered in a felicitous context (e.g., the words must be spoken with the proper intent, the pronouncer of marriage must be ordained with the ability to marry people, the partners must be willing, the pairing must be of an appropriate sort according to the bigotry of the times, etc). Another detail is that performative speech acts do more than just enact what they say, they also create something: pronouncing a marriage constructs the marriage itself, declaring war brings the war into existence, giving a promise makes the promise, sentencing someone creates the sentence, etc. Because of details like these, claiming that a particular speech act is performative says a heck of a lot more than just saying the act was performed (i.e., spoken).
On the other hand, a performance is the enactment of a particular variety of artistic expression ranging from theatrical plays, to musical opuses, to religious ceremonies, to performance art, and so on. Whether a particular act is performative is independent of whether it is (a part of) a performance. Many performative speech acts are of a ceremonial nature (e.g., marriages, divorces, christenings, declarations of war, etc) and consequently we like to make a big affair of it. Thus, these particular acts tend to be both: they're performative performances. However, many other performative speech acts are executed with little fanfare: ordering food in a restaurant, apologizing, accepting apologies, resigning from a game, etc. These are all performative acts, and yet there's absolutely no need for any sort of performance behind them. Indeed we often find it humorous, or rude, or severe, when someone chooses to turn these performative acts into performances.
The distinction between performativity and performance is crucial to understanding the thesis Butler put forth. We can expand the idea of performativity to include not just speech acts, but other acts as well. Doing so, Butler's thesis is that one's identity as a particular gender is not something which exists a priori, but rather that it is constructed by the enactment —and especially the continuous ritualistic re-enactment— of performative gender actions. The specific claim being made is that one's gender identity is an artifact whose ontological existence arises from particular deeds, in the exact same way that a marriage is an artifact arising from nuptial ceremony, that a promise is an artifact arising from the swearing of a vow, that a state of war is an artifact arising from the declaration of its existence, and so on. The performative theory of gender is often paraphrased as "gender is something we do"— but this paraphrase is grossly misleading. The paraphrase elides the entire specific content of the thesis! Sure, gender is something we do, but it's something we do in very specific ways and it is in virtue of doing those things in those ways that we come to identify with our gender. That's the thesis.
As discussed before, there are some crucial issues with performativity as a theory of gender. (Though these issues can be corrected by changing the focus without giving up the crucial insight.) But the issue with performativity has nothing whatsoever to do with the fact that performances are artificial, that performances are interruptible, that performances can be altered on whimsy, that performances can be disingenuous, that performances are "only" art, etc. Those latter complaints are why performance-based theories of gender are flat out wrong. And they're evidence of why claiming that performance-based theories were built upon performative theories grossly misconceptualizes performativity.
 Don't take my word for it, Butler herself has continually argued that performance-based theories are a gross misinterpretation of her work (Gender Trouble, xxii–xxiv; Bodies That Matter, 125–126; "Gender as Performance: An interview with Judith Butler", 32–39; Judith Butler (by Sara Salih), 62–71).