Not the most obvious advantage, possibly because a lot of Word users don't understand why this so beneficial. When producing your LaTeX document, you are concentrating on the content itself. You introduce structure explicitly by telling LaTeX when a new section begins, for example, but you don't then faff around trying to decide how the section headers should look. That's done later.
This is opposed to the average Word user, who will immediately highlight a given section header and apply formatting to it: maybe a larger font, maybe underline, etc. The point is that this will then have to be applied to every header manually. LaTeX is better as it uses a document style. This defines how different elements within your document should look (like Cascading Style Sheets defining styles in HTML pages). If you fancy a change, you only change the style definitions once, then the presentation of the document will be updated automatically. This also ensures a consistent looking document (you wouldn't believe how many stylistically inconsistent Word docs I've read!)
Word does in fact have a similar Styles feature. However, because it's optional, people don't often know it exists. LaTeX forces you to declare the document semantics (this is a Good Thing!), which is why you can rely on it to produce a consistent looking document.