Functionality
There are two places the search can scan: post title and post content. Other areas like image-related text can be searched, but those require plugins. For ordering the results, there are also two options: newer first and title first. However, the title sort takes precedence because a newer post returned from content search would display below an older post from title search. Unfortunately, some useful features haven't been included yet, even more simple ones like autocomplete and spellcheck.
URI
The URI is used as a convenient place to hold the search query. In order to hold it in the first place, you need to use the GET syntax, which isn't necessarily user-friendly if they wanted to copy the link or something, but that format can be changed. If you for some reason made the worst search form ever, or the user just wanted to edit the values outside of the site, they could use the URI for that.
Current Term
However, the most reasonable solution would be grabbing that GET value and displaying it in the search boxes. The GET also becomes the most convenient way for you to pass around and use values. It would need to be parsed as a result of being in URL-friendly format, but that would be no problem with regex.
Summary
I can see some reasons to not use Wordpress search, but I think it would be fine for most general cases. If anything, it's easy to use the searched value.