what’s the difference between system and acceptance testing?

Acceptance Testing checks the system against the "Requirements". It is similar to system testing in that the whole system is checked but the important difference is the change in focus.

System testing checks that the system that was specified has been delivered. Acceptance Testing checks that the system will deliver what was requested.

The customer should always do acceptance testing and not the developer. The customer knows what is required from the system to achieve value in the business and is the only person qualified to make that judgement. This testing is more of getting the answer for whether is the software delivered as defined by the customer. It's like getting a green flag from the customer that the software is up to the expectation and ready to be used.