Of course there are many differences between undergraduates and graduates, including:
Mathematical independence. In a graduate class, one may generally presume a much greater level of mathematical independence and self-motivation than in an undergraduate class.
Mathematical sophistication. Graduate students generally are more knowledgeable than undergraduates and will benefit from more sophisticated explanations. Graduate students are more likely to appreciate it when the instructor follows sideline topics in class for a bit, or unifies topics with ideas from other areas.
Capacity to be challenged. Graduate students have a far larger capacity to be challenged mathematically. One can give mathematical puzzles, and not be surprised to find that students struggled with them for hours and hours over the weekend.
Because of these differences, I often organize my graduate classes in a much different manner. For example, in order to introduce graduate students to the math research experience, I usually require my graduate students to write a term paper (see my account of this here), which I then collect into a "Proceedings of Graduate Set Theory" volume at the end of the semester and have a session devoted to student talks. (With undergraduates, in my experience, this doesn't work as well.) Also, in lectures I am willing to follow tangents in the topics that might come up
for a longer time and more thoroughly than I do in my undergraduate classes, since in the undergraduate classes I feel more compelled to stay closer to the established syllabus.