TITLE: Demonstrating that rigour is important
QUESTION [204 upvotes]: Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with statements that have proofs than we are with statements that lack proofs but for which the evidence is so overwhelming that it is not reasonable to doubt them.
That is not the question I am asking here, though it is definitely relevant. What I am looking for is good examples where the difference between being pretty well certain that a result is true and actually having a proof turned out to be very important, and why. I am looking for reasons that go beyond replacing 99% certainty with 100% certainty. The reason I'm asking the question is that it occurred to me that I don't have a good stock of examples myself.
The best outcome I can think of for this question, though whether it will actually happen is another matter, is that in a few months' time if somebody suggests that proofs aren't all that important one can refer them to this page for lots of convincing examples that show that they are. 
Added after 13 answers: Interestingly, the focus so far has been almost entirely on the "You can't be sure if you don't have a proof" justification of proofs. But what if a physicist were to say, "OK I can't be 100% sure, and, yes, we sometimes get it wrong. But by and large our arguments get the right answer and that's good enough for me." To counter that, we would want to use one of the other reasons, such as the "Having a proof gives more insight into the problem" justification. It would be great to see some good examples of that. (There are one or two below, but it would be good to see more.)
Further addition: It occurs to me that my question as phrased is open to misinterpretation, so I would like to have another go at asking it. I think almost all people here would agree that proofs are important: they provide a level of certainty that we value, they often (but not always) tell us not just that a theorem is true but why it is true, they often lead us towards generalizations and related results that we would not have otherwise discovered, and so on and so forth. Now imagine a situation in which somebody says, "I can't understand why you pure mathematicians are so hung up on rigour. Surely if a statement is obviously true, that's good enough." One way of countering such an argument would be to give justifications such as the ones that I've just briefly sketched. But those are a bit abstract and will not be convincing if you can't back them up with some examples. So I'm looking for some good examples.
What I hadn't spotted was that an example of a statement that was widely believed to be true but turned out to be false is, indirectly, an example of the importance of proof, and so a legitimate answer to the question as I phrased it. But I was, and am, more interested in good examples of cases where a proof of a statement that was widely believed to be true and was true gave us much more than just a certificate of truth. There are a few below. The more the merrier.

REPLY [2 votes]: Gowers is particularly interested in cases in which a "proof of a statement that was widely believed to be true and was true gave us much more than just a certificate of truth." Reading through the answers, I am struck by a recurring theme.  There are many cases where

something seemed impossible;
it was impossible;
a lot of people didn't see the point of proving impossibility;
the search for an impossibility proof led to the discovery of structures that arguably would not have been discovered otherwise.

Just to list a few examples explicitly:

 It seemed impossible to prove the parallel postulate from the other axioms. The proof of impossibility led directly to non-Euclidean geometries.
 It seemed impossible to "square the circle."  The rigorous proof (Lindemann–Weierstrass) forms the foundation of modern transcendental number theory.
 It seemed impossible to solve all polynomial equations using radicals.  Rigorous investigation of this impossibility led to what we now call Galois theory.
 (Daniel Moscovich's answer) It seemed impossible to unknot a trefoil.  The search for a rigorous proof led to the discovery of all kinds of knot invariants.  More generally, topology and geometry are rife with examples of "obviously inequivalent" structures, and the search for rigorous proofs has uncovered all kinds of important invariants.
It seemed impossible to write down a procedure that would mechanically determine the truth or falsity of an arbitrary mathematical question. The rigorous proofs (incompleteness/undecidability) revealed fundamental limits to human knowledge.

In some cases, you could perhaps quibble with my claim that most if not all people thought these things were impossible, or that people didn't see the point of trying to prove impossibility.  In this regard, let me mention the article Why was Wantzel overlooked for a century? The changing importance of an impossibility result by Jesper Lützen. Lützen persuasively argues that Wantzel's proofs of the impossibility of trisecting the angle and duplicating the cube were all but ignored because people just weren't that interested in an impossibility proof of something that everybody already believed, or suspected, was impossible.  So even mathematicians are not immune to the tendency to undervalue negative results.