Just to play devil’s advocate… ;-)
I agree wholeheartedly with the above responses; I generally don’t see any advantages to not having a button, at least from a UX perspective. Having this button present meets the user’s expectations, and is often an absolute requirement to support individuals who aren’t tech-savvy.
But, I feel it's quite important to consider that the presence of a button can cause problems if it isn’t approached properly.
I ran usability testing project for an e-commerce site not too long ago, and the ‘search’ button was causing a rather nasty usability problem. As the button had been made very prominent in the design, particularly in contrast to the text entry field itself (which hadn't been well defined), many users were clicking the ‘search’ button under the assumption that it would open or display the site search, as oppose to initiate it.
As a result, many users ended up a “0 results found for (null)” page and became quite confused. We ended up rectifying this problem by making the text field more prominent, downplaying the button and changing the terminology to ‘go’ as opposed to 'search'. This rprevented and confusion or ambiguity as a result of the button. The decision to include a button within your search field is probably the right one, just ensure that it’s fully considered in terms of design or it can end up doing more harm than good.
That's my 2 cents anyway :-)
