Whereas the Mandelbrot set and other non-Julia fractal sets are defined entirely by the their coordinates (which specify its location in the complex plane), the Julia set needs an additional parameter -- a seed point in the complex plane.
The location of the seed point turns out to be very important in determining the Julia set's appearance. In fact, the Mandelbrot set was first discovered when Benoit Mandelbrot was plotting how different seed locations affected the Julia set.
If the seed location is in the Mandelbrot set (the black area is the only part which is the Mandelbrot set -- the beautiful coloured bands around it are merely very attractive side effects), the Julia will seem "connected." If the Julia seed is taken from outside the Mandelbrot set, the Julia is "dust" or Cantor Dust, to be more precise.
These two different types of Julia sets -- connected and dust -- have technical definitions that involve a lot of mathematical terms used in their most precise and obscure ways. However, the basic idea is that if a Julia set is connected, it looks connected, if it doesn't, it's dust. If the center point of the Julia set is black, then it's connected, if not, it's not. The Mandelbrot set is the set of all seeds for connected Julias.
The relation between the Mandelbrot set and the Julia seed location doesn't end with the rule stated above. If you put the Julia seed in the medium-sized bulb at the top of the very largest bulb, you will notice that the Julia set created is made up of black areas that always meet together in threes. As you move around the bulb, the exact shape changes, but that rule remains the same. If you move down and to the right, to the next largest bulb in that area, you should see the Julia set change so that the black areas now meet in fours. At the next largest bulb, they meet in fives, and so on. Now go back to those bulbs in the Mandelbrot set and zoom and scroll with the mouse towards the main arms coming out of the ends of each of those bulbs. You will notice that the arms meet three at a time, then four at a time, then five at a time...