There is only a slight difference.
All search engines understand parameters. Parameters have been around since the beginning of search engines.
There is only one difference from an SEO perspective and it will not help you much.
Using your example, Google will remove all special characters to obtain the terms within the URI.
?s=ajax becomes ajax
search/ajax becomes search ajax
In this particular case, you have one more term for search, however, that additional term
search, is such a common word it will have little to no value. But because search engines can recognize this is a search page, and having the additional term
search is a valuable clue, it will likely take note of
ajax as being an important term for the page. If your
search/ajax would be relatively equal.
But what is important is that you likely do not want your search pages indexed. You want the content pages indexed. Search engines do not generally like search pages since in the past they were used to spam search engines.
Instead, create a sitemap that has all of the content pages you want search engines to know about and restrict access to your search function. You will get much better results.