schema.org is only a collection of vocabularies (primarily, but not exclusively, for use with microdata), so this question should be: microdata vs. microformats. And why not invite RDFa to the party?
RDFa and microdata are not the same, but similar. microformats however differs from both strongly.
If your only aim would be to enhance the display of search results from search engines, it doesn't matter which markup way you choose (as long as it is supported). But "semantic markup", of course, allows much more: building the Semantic Web. Not without reason microformats relates to the term "lower-case semantic web", while RDFa relates to "upper-case Semantic Web" (microdata is much newer, but it would fit into the upper-case variant). The main difference: extensibility. RDFa and microdata use URIs, microformats uses pre-defined class names. With microformats you can only mark-up certain content, if the community created and accepted an appropriate "vocabulary". With RDFa and microdata you can create your own vocabulary, if there doesn't already exist an appropriate one.
In other words: RDFa and microdata define a way to semantically mark-up content, but they don't supply vocabularies. microformats defines a way to semantically mark-up content and (solely) supplies vocabularies.
Google says I can't use both "because it may confuse the parser".
I wouldn't let this stop me from implementing several markup ways. Also, Google kind of revoked this statement in a chat.
See also the questions: