To iterate is human
Iterator versus the Enumeration Method
As a distinct pattern, Enumeration Method was first properly documented by Kent Beck in Smalltalk Best Practice Patterns.
However, it is not a pattern that is restricted to Smalltalk: it can be applied in C, using function pointers, such as EnumChildWindows in the Windows API; it can be used in Java, based on the common Command pattern and the specific use of inner classes to achieve a sense of closure; it is the common form of iteration in Ruby, which supports blocks as objects directly, and where these are commonly (but confusingly, for our purposes) also known as iterators; for the functional programmers among you, it is in essence the map function from an object-centred perspective.
Of course, the ease with which Enumeration Method can be implemented and used, and therefore its applicability, is an important consideration. It would be too simple to claim that Iterators are necessarily more verbose than Enumeration Methods and that Enumeration Methods are generally superior, for the simple reason that such a claim needs to be made and measured against a specific context. Understanding the role context plays in design is perhaps one of the most important, but most overlooked, aspects of successful pattern application.
In a language where blocks are supported natively as objects, such as Smalltalk and Ruby, implementing Iterators without appropriate cause might be considered quite curious and more than a little gratuitous. However, although it is said that "what's good for the goose is good for the gander", it doesn't follow that it's always such good source for the mongoose.
In a language that doesn't support closures, it turns out that even though implementing the Enumeration Method itself is normally easy, using it can be something of a pain, shifting the complexity from the collection writer to the collection user.
This applies to a greater or lesser degree depending on what other features a language supports and what its native library style is. For example, assuming that closures are adopted in Java 6, Enumeration Method will become easier to implement and use in Java. For the moment, however, although anonymous inner classes make a block-like approach possible, the resulting syntactic overhead is somewhat cumbersome if you aren't getting any obvious additional benefit.
So unless there is a specific reason to do otherwise, such as recursive traversal or synchronized traversal, it is far wiser to favour Iterator as the default approach in Java: both the language and the library are geared up to support it, and writing an Iterator correctly is not a significant challenge.
We could go on to talk about Python's approach to iteration, or the diversity of styles that can be conveniently supported in C# 2.0, or the style of iteration used in C++ that supports the concept of generic programming, or understand how simple and effective map is in Scheme, or the relationship between iteration in Ruby and CLU, and so on. But, for the sake of brevity, I'll stop the language listing there. Hopefully you get the general idea: there is no single option that is best across all languages.
So, whatever its merits elsewhere, if a particular pattern cuts across the grain of a language's features and its received idioms, it is normally easier to go with the flow than against it. You don't want to be writing code that is unnecessarily complex by virtue of an unquestioned idiom import from elsewhere; an idiom that fails to add any noticeable advantage over the more native idiom for a typical case of application.
But let's be clear: that is "normally easier", not "always easier". It pays to be a polyglot: you want your design gestalt to contain more than just a nice set of unquestioned defaults informed only by a single language. To be able to select the mot juste, whatever the situation, you want your design vocabulary to be able to draw on multiple sources. You need more than one idea.
I've already mentioned that for recursive data structures Enumeration Method is much easier to implement than Iterator. Likewise, where you have a collection shared between threads and you want to support uninterrupted traversals, it is much easier to use.
These situations sound quite specific, but having explained why Enumeration Method is a less appropriate approach than Iterator for most uses of iteration in certain of languages, I would like to ensure that not too much design territory is ceded.