Some projects get the final acceptance but just a few weeks later nobody is using them. Why is that? And why should the project manager care?

The real issue relates to stakeholders' expectations, vs. blindly applying the "iron triangle" to manage a project. For years I'm advocating that we are applying the wrong definition for "project success" using the triple constraint - and your example is a clear example of failure while meeting the traditional definition of "success". Moreover, let's take a look at the CHAOS reports - the success/failure/challenged levels are roughly unchanged for many years, so are we really not learning anything?
To me, a project is only as successful as the stakeholders think it is! It does not matter if the project was couple of weeks late, or costed a bit more than the initial estimate (we're not prophets after all), or did not deliver every "i" and "t" - if the stakeholders are happy with the outcome the project is acclaimed as a success. If not, all the signatures in the world will not make people actually use it...

Was organizational change management part of the project scope or, at least, done alongside the project? If not, the end users may not have been prepared and ready for the product(s) issues from the project.