Me and my DH started out with the Billings method but found it difficult to learn. Part of this was certainly due to communication problems between myself and the instructor. After a few frustrating months, we switched to Creighton. The main difference was that Creighton method provided very clear, specific things to watch for with the CM, and a very specific way to check for CM. Billings requires the woman to interpret it based on general guidelines, and I found it was too difficult for the instructor (and me) to interpret my fertility, even after 3 months of charting. After switching to CrMS, I was immediately more confident with my charting.
The main drawback with CrMS versus Billings is how strict you need to be in the manner you take the CM observations. It does take some getting used to, to build the habit. I could see a woman preferring Billings if she was more comfortable using her own interpretations and observation method. But my confidence level in CrMS is way, way higher. We did have to pay again for the materials and go through the training again, but our CrMS instructor is fantastic, and got us through the material as quick as possible. Oh, also, I've been told that Billings has better smartphone aps than CrMS!
The main drawback with CrMS overall is the abstinence during fertile times Of course you can choose to use a barrier method, but there are strong cautions about higher pregnancy rates when using condoms (etc) instead of abstaining during fertile days. I personally haven't used condoms during my fertile days while charting, yet. But for the past few years my DH and I used condoms exclusively, without charting, and certainly during my fertile times, and we never had any issues.
Another thing to be aware of is that both Billings and CrMS come from a Christian background. If you're not Christian, you can certainly still learn the techniques for either method and use it for effective fertility planning. If you are, you may find the extra dimension of religion a welcome source of support as you learn the process with your partner.
That's my 2 cents End of the day, I've been charting CrMS for the last 2 cycles and I can see myself using it for years.