On MathOverflow, Timothy Gowers asked a question titled "Demonstrating that rigour is important". Most of the discussion there was about cases showing the importance of proof, which people on CSTheory probably do not need to be convinced about. In my experience proofs need to be more rigorous in theoretical computer science than in many parts of continuous mathematics, because our intuition so often turns out to be wrong for discrete structures, and because the drive to create implementations encourages more detailed arguments. A mathematician may be content with an existence proof, but a theoretical computer scientist will usually try to find a constructive proof. The Lovász Local Lemma is a nice example .
I would therefore like to know
are there specific examples in theoretical computer science where a rigorous proof of a believed-to-be-true statement has led to new insight into the nature of the underlying problem?
A recent example that is not directly from algorithms and complexity theory is proof-theoretic synthesis, the automatic derivation of correct and efficient algorithms from pre- and post-conditions .
-  Robin A. Moser and Gábor Tardos, A Constructive Proof of the General Lovász Local Lemma, JACM 57, article 11, 2010. http://doi.acm.org/10.1145/1667053.1667060
-  Saurabh Srivastava, Sumit Gulwani, and Jeffrey S. Foster, From program verification to program synthesis, ACM SIGPLAN Notices 45, 313–326, 2010. http://doi.acm.org/10.1145/1707801.1706337
Edit: The kind of answer I had in mind is like those by Scott and matus. As Kaveh suggested, this is a triple of something people wanted to prove (but which wasn't necessarily unexpected by "physics", "handwaving", or "intuitive" arguments), a proof, and consequences for the "underlying problem" that followed from that proof that weren't anticipated (perhaps creating a proof required unexpected new ideas, or naturally leads to an algorithm, or changed the way we think about the area). Techniques developed while developing proofs are the building blocks of theoretical computer science, so to retain the value of this somewhat subjective question, it would be worth focusing on personal experience, such as provided by Scott, or an argument that is backed up by references, as matus did. Moreover, I'm trying to avoid arguments about whether something qualifies or not; unfortunately the nature of the question may be intrinsically problematic.
We already have a question about "surprising" results in complexity: Surprising Results in Complexity (Not on the Complexity Blog List) so ideally I am looking for answers that focus on the value of rigorous proof, not necessarily the size of the breakthrough.