"Customary law" seems like a contradiction in terms. Custom is the way people live and the traditions they follow. Custom gets tested and reviewed with every conflict. The members of the community (or elders, or whoever customarily decides) have to get together and decide if so-and-so did the right thing in such and such a situation and then determine what would be a fair resolution. A fair resolution may involve trampling the offender into a bog, of course. The offense, the judgement, and the resolution all happen within the community. No one pulls out a book or list of decisions.
Law is a written set of rules that a state enforces on people. Sometimes there is a codified book of rules, and sometimes there is a book of previous decisions that other state law enforcers have made. Custom isn't relevant, unless the state enforcer decides to consider it relevant.
For most of their history, the Romans limited the scope of law to affairs that directly impacted the state. They left everything else to custom. Even murders within families were formally considered matters for the family to resolve. Parricide was an exception, since it upset the familial order that the state relied on.