How do graduate research methods differ from those learned in undergraduate research?
The biggest difference between undergraduate studies and graduate research is that graduate research very specialized; in many fields, the terms used by graduate (PhD) researchers will not be familiar to most undergraduates.

As to whether the methods are different or fundamentally the same, this is a matter of philosophy. Certainly, the methods in graduate school build on those you learn during your undergraduate, and it is necessary to be an expert on the undergraduate methods before becoming a good PhD student. Depending on the field, that may be basically enough to be competent for graduate research, or, you may have to learn entirely new ways of thinking. For example, in mathematics, graduate research requires thinking at a higher level than the detailed writing of rigorous proofs in undergraduate courses.