CNMs get a regular nursing degree and then go through a college training program to specialize in midwifery. CNMs may work at a hospital, bc or at home, depending on regulations for the particular state they live in.
CPMs are certified through NARM, a certification-process established by the Midwife Alliance of North America (MANA), a professional organization for non-nurse midwives. CPMs learn the trade through study and apprenticeship with an experienced midwive. They also have to take classes and get training in various things, like neonatal resuscitation.
CPMs learn the trade in homebirth settings through direct experience, whereas CNMs learn in a hospital setting, under the supervision of OBs and other midwives. So, by training, CPNs are the ones more "steeped" in natural birth from the beginning. However, that doesn't tell you anything about an individual practitioner. CNMs can also be very naturally minded... especially those bucking the system by practicing in out-of-hospital settings. They may also have the added advantage of being able to continue your care in the hospital if a transfer is necessary (this will depend on the state and your individual mw).
Not all states recognize the CPM credential. In some states, CPMs are licensed and practice according to the regulations in their states. In other states, it is illegal for any non-nurse midwife to attend a birth. In many other states, CPMs are not licensed, but are not forbidden to practice by a direct law. So, they have fewer regulations to deal with, but there is always the fear that if something goes wrong they could be prosecuted for practicing medicine without a license, and end up losing their livelihood and their freedoms.
There is also a newer certification called CM, or certified-midwife. This was introduced by the CNM group to credential midwives through a more direct-entry rather than nursing approach. There are very few CMs in the country, and I think the only state to officially license them is NY (although I may be wrong).
When choosing a provider for a homebirth, I would look at the specific qualifications and attitudes of the individual, rather than their credential. It's most important that you are comfortable with the mw and your personalities mesh. You will also want to know up-front what policies and regulations your mws follow. At what point do they risk people out? Do they do VBACs, breeches, or post-42-weeks births? Some of these things may be determined by state law, and others by the mw's individual policies, but you will want to know what they are up front. There also may be a difference in what services they can provide in terms of ordering bloodwork or ultrasounds, etc. That varies from state-to-state based on licensing regulations.