Japanese Mochi / Korean Mozzi - Same thing?

Well, the question is basically in the title... I've wanted to try mochi-sweets (daifuku?) ever since I know what they are, but all my asian market cares are korean rice-things with read bean filling called mozzi. Is it the same thing? They are a little wobbly on the outside and come in milky white or green-ish colors. Hard to explain... Here is a picture of one. They come in different sizes and are generally very sweet, hard to compare to any western food in texture and, well, way too expensive for their size actually.