like all other things, unless the question is more specific, it all depends on situation. small numbers of results are emanable to live search and result display. what is this searching for, text in a database index, image based results, etc? At the crux of it the button is needed when there might be a delay in producing results or otherwise livesearch is not possible. the button helps users to connect to an action (search) and can provide cues about progress (disabled, progress bar etc.) Without such a "start" mechanism what is any progress (if even viable without a button) relative to but that action of clicking the button. there's also a slightly deeper need for users to connect to a search action, the instigation of the search and result retrieval. on the purely visual side of things, the button is also a cue that the search field is not something else entirely. some contraptions can assist in providing such cue but the simple button does so unequivocally without raising a sweat.

Unless the question is more specific, a simple answer is not possible. If the search yields a small number of results, then likely (but not necessarily) it would allow a live search and result interface. As you get results updated each keystroke, there's probably no need for a search button.

What is this searching for? Is it searching text in a database index (e.g. a web index), searching a library of images, etc? What the specific context is should be taken into serious consideration before a decision is made to exclude a search button. I have seen too many people of the old and new generation interface design fall into the same trap of going through the motions of design totally blinded by preconceptions.

On a separate note. When a search button is present, it helps users to connect mentally to the action of instigating a search. Besides the mind and action connection, the search button is a valueable interface element to provide cues about search progress. The button can be disabled showing that a search is underway, and the interface can show a progress bar if the search usually takes a few seconds or more. Without this clear and unambiguous time reference as pressing a search button these affordances aren't viable. Using the enter key by itself without a button in this way is nowhere near as effective.

On the purely visual side of things, the button is also a cue that the search field is not something else entirely. So the search button indirectly labels your text entry element as a search field. Some contraptions (e.g. image icons) can assist in providing such cue but the simple button does so unequivocally without having to devise other cues.