Yes, identical twins are due to a single fertilised egg splitting into two separate eggs before they each develop separately. The twins come from the same original egg and sperm. They are thus genetically identical; they are naturally occurring clones. Fraternal twins are when two eggs happen to ovulate at once and both end up being fertilised. The twins are genetically not the same (since they came from different eggs and different sperms), and are nothing more than ordinary siblings.
Since gender is determined by DNA, identical twins will always have the same gender, barring a sex-change. Fraternal twins can be whatever, just like any other pair of ordinary siblings.
How about: the difference between astrology and astronomy. I was teaching a class about the solar system, and a disturbingly large number of the kids' parents thought, based on the unit name, that we would be learning about gemini and sagittarius and all that. Some rang up to ask why we would be spending so long (4 weeks) on it.