I would agree that I don't think there is not an actual defined difference.
Users and customers are often the same people, and sometimes they are only one of those two things:
1: User journey- people aren't necessarily using an e-commerce site, it would be a bit odd to call users on the health service's website a 'customer' for example.
2: Customer journey- the focus isn't necessarily on the systems. They could be looked at in a very abstract way if they exist at all and the focus is more on the customer's general experience in a real world shop.
Circumstances also come to mind where there would be a separate customer and user journey, such as is in a system that is designed for e.g. customer service reps, staff working in a shop, etc... For this kind of system it could be rather important to have a separate customer journey and user (staff) journey.
Overall - user journey is the more usual word when you are looking at a particular system. There is absolutely no harm in replacing user with a more 'friendly' word like customer, employee, Bob's, etc....A different name doesn't alter what it is.