Both of them have been around for ages. Mostly they have been used for the common purpose of decorating the windows and making the interiors look good. Drapes were the cloths that were hung in front of the walls. The drapes were more in use by the rich and the curtains were mostly used in different types and were used by all and sundry. Curtains were at one point in time defined as a wall that is used to enclose a certain area such as a barn yard. Therefore, back in the day, curtains were not necessarily made of cloth.
There is a difference in the types. The drapes are generally made from heavy fabric that is often textured and not only does it provide some visual delight, it also helps in protecting the indoors from the severity of sun or blowing winds. The curtains are not necessarily heavy fabrics and are often made of light weight cloth which generally lacks the same beauty of the drapes. The idea for them is also to shield the indoors from extremity of the weather as well as onlookers.
There is a difference in price as well. Because of the nature of the product, the drapes can cost quite a bit. Even a simple drape will be fairly priced. On the other hand, the curtains can be pretty economical but there are those that cost quite a bit. There is a great price variation and you can get curtains for just about any budget that you have at hand.
Drapes are more formal in nature. The mechanism used to operate the drapes is often hidden behind it. On the other hand, the mechanism through which a curtain can be moved is in front of it and can be seen. The rod with which the curtains are hanging is actually a part of its decoration and makes it look good as well.