CPMs are Certified Professional Midwives. They have a certificate, not from a college, but from a midwifery program geared towards natural birth.
Whereas CNMs typically have a degree from an accredited school (usually a masters degree from what I've seen). CNMs have to carry malpractice insurance like doctors do, and the vast, vast majority only deliver in hospitals/birthing centers. CPMs only do if it's required by law in the state they practice midwifery. And even so, most won't risk taking on high risk deliveries (though you can find a small percentage who will deliver twins or breech babies... but I find that more in states where midwifery is less-regulated), and they have no hospital privileges. I wanted to hire a homebirth midwife this time around but I had complications (a hemorrhage) in the first trimester that risked me out of a homebirth, so I'm delivering in a hospital instead. (Then came the diabetes dx... which would've risked me out had I not had the hemorrhage anyway)