Here is what you need to know.
Assuming for a moment that both sites are equal...
The Google Index contains, at minimum, the domain name, the full URL, any link with a
to relation (think relational database) to the URL, the HTML source, and analysis of the content. While there is quite a bit of conjecture over what analysis exists over the content, it is at least known that Google breaks down the HTML into the HTML DOM elements and each one is assigned an ID that identifies the order of the DOM elements, any parent-child relationship, any dependency of one element upon another, and the type of element. With that, content blocks are determined. For example, the entire content following the h1 tag is one block, any header tag is a block, the content following a header tag is a block, each paragraph is a block, etc. Each term within a block is given a position in term count from the beginning of each block and associated with a term in the term index.
This is important for two reasons. One, the content can be understood without the original HTML. And two, each part of the content can be analyzed in a variety of ways including linguistic semantics, topical semantics, and many other forms of analysis.
Why did I tell you this?
Two reasons. One, because this is the basic form of analysis that Google and other search engines use. And two, domain names (URL), link text, URI (path and file name separately) are also analyzed in this way.
Each content block and DOM element can be analyzed using semantics, primarily for topic and PoS (parts of speech - not the other one) relations in understanding the content itself, and in raw term match. The primary matching mechanism between the search query and the content is topic and term matches. Between the two are ontologies that allow many forms of analysis to know that a car is an automobile, that Rockefeller Center consists of 19 addresses, that Babe Ruth is a person or a candy bar.
Keep in mind that Google does not do direct term matches, even if that is what it appears to do.
As each term within the content is analyzed, matrices are created to score the term according to several analysis methods including semantics. Add to that score is weight for the element as it is seen in importance. This is how a term in one thing can outweigh the same term in another thing, for example the title tag versus the h1 tag. This list is the approximate order of importance that Google assigns weight to.
- The domain name.
- The URI path.
- The title tag.
- The link text.
- The description meta-link.
- The content itself.
- The header tags.
- The URI file name.
All things being equal, using a domain name of
paper airplanes and a URI path of
paper airplanes, the domain name will score higher. This is because Google believes that semantic clues in the domain name are more important/significant than in the URI path. By the same token, a URI path of
paper airplanes will out perform a title tag of
Your question is far more complicated than that however.
What is not considered are other important elements such as the title tag, the description meta-tag, and links to the page, etc., all of which are very significant in adding weight to the semantic analysis.
How one will perform against another cannot simply be domain name versus URI because there are so many factors that can easily out weigh the two, however, if you were to break it down to the simplest understanding the domain name will out perform the URI path every time. That is, until Google changes it's mind.