Many sites now provide a search bar without the buttons, and everyone (?) understands that you press enter afterwards anyway.
Is it beneficial to keep the Search button for this case? Do I gain or lose anything by removing it?
This answer is nearly a duplicate of a question answered earlier today, but I don't know how to close and redirect your question.
For a long time, now, these have been the Nielssen Norman Group's published guidelines (available for purchase, not free). They sum up as follows:
- Have an easily identifiable search box in the upper right-hand corner of the page, with an open-text field accompanied by a Search button.
- The search box needs no label. A clear Search button next to the field identifies the search for the user and tells them how to execute the search.
A button helps people recognize that there's an additional step to trigger the search action—even if they decide to do this by pressing Enter.
Versions of this icon are increasingly common. For now, NN/g recommends retaining the button, and has also provided some supplemental recommendations to help make search more usable for your users. For example, pay attention to the placement of the search box.
Given your illustration, above, you might also want to check out the recommendation against putting instructions inside the search box in the same article.
Hope that helps you move things forward.