A classic example of a computable enumeration is to generate all valid statements. A classic example of a problem that is computationally complementary computably enumerable is to prove a statement true. The reason why the two problems are different is that not all statements are either true or false (valid or unsatisfyable). Only after infinity could we be sure that a statement is unprovable (invalid but satisfyable). 

Every problem that is computationally complementary computably enumerable is what is 'being left over' from a computable enumeration. The reason that this is different from a set complement is that the elements that will not be enumerated are combined with the elements that are yet to be enumerated. That is why I prefer to call computationally complementary computably enumerable as computably approximatizable. (I find them reminiscent of Cauchy sequences.) 

Good examples of computable approximatizability as well as the relationship between computability and computable enumerability are numbers and numerals. Numerals are a kind of string. Numerals are an unambiguous representation of numbers. Numbers are an abstract data structure. 

The Nth digit of a numeral or upto and including the Nth digit can be computed. To get the entire numeral, it is often necessary, to computably enumerate it (even for rational numbers). Transcendental numerals can be computably approximatizable. 

Total computable functions are important to complexity theory. Every total computable function has a corresponding computable enumeration. To form a computable enumeration procedure combine a total computable function with the succesor function and repetition. In other words, mapping a total computable function over a seqence of numerals of the natural numbers results in the output of an associated computable enumeration.