The Basic Checks list instances where a server or client is violating the HTTP protocol in some way. Such violations are often harmless and occur unnoticed, but they may cause clients or servers to fail if they strictly adhere to standards. Or, the violation may reflect a bug or unexpected codepath in the client or server that should be fixed.
The Extended Checks list instances where the client or server is doing something which might cause a problem for a client. For instance, Fiddler's extended checks will warn you if you try to use Set-Cookie to set a cookie containing an unescaped comma. Most browsers don't mind such cookies, but clients which strictly adhere to the specification for cookies (e.g. the .NET CookieCollection class) will reject such cookies.