Even in a commercial context they aren't the same. The user is who sits in front of the screen, and the customer is who you have to please to make money.
You're right. But, nowadays these terms get more context meaning. For example - a user is someone who platforms sell their data to customers. Or a user is someone gullible you trick into making decisions that are good for the business.
The literal term of someone who is using your system/product has evolved.