According to posts on Apple's support forums and Twitter, many of those who have upgraded to iOS 6 are experiencing Wi-Fi connectivity issues. Based on the claims, the issue has arisen from a network verification process baked into the software, wherein the OS attempts to load a dummy page on Apple.com to detect if a users is connected to a paywalled network. Unfortunately, however, the verification page seems to lead to a 404 error, resulting in an inability to connect to Wi-Fi for many users.
Update: Apple quickly rectified the problem by updating the page to remove the 404 error. Users should have no further issues connecting. However, the problem does raise concerns about the verification method and its impact on Wi-Fi-dependent iOS products, should Apple.com ever experience a lengthy outage.