CX is very much product related, and UX is more brand relevant.
Traditional UX is a relatively small and tactical component of CX. CX is focused on a company's vision of success for their customer. CX requires definition of a customer's desired outcome and success criteria when engaging products, services and support. The practice of CX requires competence in business process optimization, product strategy, service delivery design and strategic planning. UX requirements for interfaces and systems in support of the CX are downstream of the CX design process. UX teams are really stabbing in the dark if they aren't designing within the constructs of a CX-customer experience vision and design. However, good UX teams do this instinctively, even if unknowingly, as part of their creative process.
It's a check and balance system - UX is there to framework the interactions with products or services while CX's purpose is to maintain the relationship with the customers. But, what is often missing is some sort of formal methodology to verify the UX strategy and designs against the CX vision and design. And in the case that a company has a formal CX focus, the UX team should be integrated with the CX team. A company who wants to embark on true CX improvement would probably be best served by adopting UX methodology as the foundation of their CX process, and UX strategists as leaders of the CX team.
UX is a part of the CX. The "thing" that is created and delivered to a user does in fact impact the users "experience", certainly it does not encompass the entire users experience as a user's overall experience is a product of their current environment, past experiences, etc. The one significant difference between a user's and customer's experience is that a customer would have provided some compensation to the provider to delivering the "thing". In the present this compensation could be monetary or could be taking the time and resources to download a free app. The expectations of a customer versus a user are different based on the compensation provided. Hence, a good UX designer should be considering the presale environment, use environment, and after sale environment. How a user engages and communicates through a customer service experience is part of their association with the brand. Knowing the backend process for a CX representative allows for there to be collateral material on the front end to anticipate issues and point them to the correct direction when they have a problem in their account.
From a conceptual standpoint, CX always trumps UX in order to uphold what drives the bottom line. However, in many organizations, there wasn't a good integration of the UX strategy with the business strategy. And it means that there wasn't a clear vision, and people weren't asking all of the right questions early in the process. Isn't it the skills and responsibility of the UX designer to control their own destiny when it comes to establishing what will and what won't work within the overall business strategy. Over time, you gain those skills and learn how to collaborate with business stakeholders, even if they don't have a clear view of their customer. And so in the fact that a truly great UX professional is the one who has the ability to integrate with and fully support the desired customer experience, whether the business can articulate that or not. So at that point, UX and CX are converged as the same professionals, to design and optimize overall customer experience.
UX is very much product related -how users interact with the products or services, CX is more brand relevant -how people think about the company and how they perceive the brand. And UX has to be integral part of CX, as UX for UX sake may make improvements to the interaction model with systems, but it doesn't solve core issues of engagement between the provider and its customers. And there needs to be a clear business strategy to bring the two together, to deliver delightful customer experience and build outside-in digital business with strong brand image.