What's the difference in meaning between saying "I'm an active person." and "I'm a busy person."?
So, you would say you are an active person if you do a whole bunch of things and seldom get time to just sit back and relax or do nothing.You'd say you are a busy person if you have a whole bunch of activities that occupy your time and self.
So, if you are reading a book, or just sitting on your couch pondering over things, you can say you are busy. But you would restrain yourself from saying you are active while you are just sitting on the couch thinking stuff. This example suggests towards the meaning above: being active usually corresponds to being engaged in physical activities.