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. [1]

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.