I don't mean wave drag here. What is compressibility drag, which is understood to be a form of miscellaneous drag?
Let's do a Gedankenexperiment:
Think of air streaming around a body as flowing inside a stack of flexible tubes. The walls of the tubes are impenetrable, infinitesimally thin and follow the local streamlines faithfully. When the body approaches at subsonic speed, the air in the tubes near that body makes way for it by speeding up: This reduces the needed cross section and lowers static pressure, so the total pressure will stay constant. On the back side of the body the air slows down again and the tubes regain their old cross section and static pressure. Bernoulli in action.
When the speed nears the speed of sound, however, the speeding up is joined by a drop in density. Still, the air near the body speeds up but that will not change the cross section as much as before, because now this speed increase is coupled with a loss of density. The cross section still drops, but not by as much as before. More tubes have to bend away from the body and need the air in them to speed up so the body can squeeze through. More general: A change in body thickness (more precisely: The second derivative of its cross section) will work on more tubes, so its effects do not die down as quickly as in subsonic speed.
At the speed of sound the cross section decrease due to speed changes is exactly balanced by the drop in density, so the same mass of air needs more volume and eats up all the gain from increased speed. Now there is a wall of air which cannot yield facing the approaching body. That is the sound barrier. In reality, the speed around that body does not reach the speed of sound at the same station in all tubes, so there are mildly sub- and supersonic sections which will allow it to squeeze through. Still, drag is much increased and depends heavily on details in the body contour.
At supersonic speed density changes more than speed, so in order to reduce its cross section, the air in the tubes will slow down in order to make way for the body. Since it has no advance warning of the approaching body, it does so in a shock. As a consequence, the cross section can now be reduced because density increases in that slower air past the shock. Static pressure increases also so total pressure can stay constant again.
This thought experiment was explained in 1951 to researchers at NACA Langley by Adolf Busemann. One person in the audience, a young fellow named Richard Whitcomb, used the insight he gained to formulate the area rule a few weeks later.
Compressibility drag is a type of parasite drag caused by the compression of air ahead of an aircraft traveling at high speed. An aircraft not designed for supersonic flight will experience it as it approaches Mach 1. The effects are noticeable once the aircraft reaches a Mach number of 0.6 to 0.7 and the coefficient of drag rises by 0.005. In subsonic aircraft design it is also considered the limit of normal economic operation of the aircraft.
Unfortunately, the definition of these two terms is not consistent across literature. Often, they are both used to describe the same effect: the increase in drag due to the presence of shockwaves.
However, sometimes a differentiation is made between the terms, depending on the way that the total drag is decomposed. You may find that compressibility drag is used to describe the increase in drag due to an increase in mach number at constant lift (thus a composition into zero-lift, lift-dependent and compressibility drag), whereas wave drag is used for the drag which is "physically" caused by the presence of shock waves.
In that case, the values can differ. Take for instance a certain flight condition at transonic speed and increase your angle of attack while keeping your mach number constant. Compressibility drag (according to this definition) then stays constant, whereas wave drag will increase. Check out the following document for some more clarification: http://mail.tku.edu.tw/095980/drag.pdf