Actually to clarify, there is an attribute type = "search" but this was introduced with HTML5. Not all browsers support it. This is why you see a difference. If the browser supports it, browser styling will apply.
If you are interested in more examples or a more detailed explanation, take a look at this overview: HTML5 Form Input Elements. Aside from the examples, there is a browser validation that will test the different input element types against the browser that is used to visit the page.