I've used both for my births, and have looked into both paths a bit on my own journey.
CPMs: You know they have met a standard of training to attain their CPM. In Oregon, they can also accept Medicaid, so more women can afford them. They can legally carry equipment that may be needed at a birth like Pitocin to help a sluggish placenta out. They also must follow state guidelines as to how they practice, for instance, if a woman goes past 42 weeks, they must refer them to an OB. If they don't, they are operating outside the scope of their practice and they may run the risk of losing their license if they are reported. Other states may mandate that they don't do VBAC, multiples or breech. The CPMs I have worked with have been very competent and caring. They knew what to do when they needed to take matters into their hands and when they could sit back.
Lay midwives: You need to ascertain whether their level of expertise yourself. They are not constrained to all the rules because they have no license to lose. At the same time, they should never attempt something they are not trained for... for instance, if the are not trained in suturing, they should tell a woman that before the birth, so if she does tear, she knows she may need to transfer for repair. An example is the last midwife I had... she was a lay midwife. She told me upfront that she did not suture. She did not do breech, although she knew a licensed midwife who was trained to do them that she could refer me to if needed.What I really liked is she was not under any time constraints. She trusted my own observations during my pregnancy, and did not need to do x number of vaginal exams for her paperwork during labor. She had the freedom to rely on observation rather than interruptions.
As a future practitioner, most midwives that want to attend homebirths seem to be following the CPM path. It is straightforward, and helps them know when they are considered ready to practice, as well as giving them legality in many places. Becoming a lay midwife is a bit less clear, not only because you are the one that must determine when you are ready to practice, but because they seem to be the target of every group to illustrate the stereotype of the illiterate, uneducated midwife. I did not find that to be anywhere near the case with the one I have worked with, and I know of others who have worked in my area in the past who are very knowledgeable and well-trained. They just chose not to have to work within the constraints of the state guidelines.
I do think it is imperative that any midwife, no matter what letters are behind her name, should be straightforward with their clients and let them know what they are comfortable handling.