An interesting question from eBizQ today. Interesting because it asked BPM or CRM.
Customer service is at its heart a process supported by an application.
So you should use (BPM: process mapping) to establish what the process should feel like from a customer’s perspective (especially now self-service is prevalent), and then you should use CRM or BPM (automation) to provide the support and you should deliver both the description of the processes and the application to the customer service agents (who may be your staff or your customers).
A recent blog called Your customers hate it, your staff hate it, but you make the do it anyway describes the issue in more detail.
But a picture tells a thousand words from Just a Thought