There are many differences and they exist because the standards are written by different committees that are focused on different parts of the industry. To get a full definition of the differences and how they pertain to a particular valve type, you need to get a copy of both, read and compare.
A short summary of some of the differences:
API-608 is ball valves only. 6D is ball, gate, check, etc.
API-608 covers up to NPS 20 size, 6D up to NPS 60.
API-608 has up to class 600 flanged, and class 800 thread and weld end. 6D goes much higher class ratings
608 requires conformance with ASME B16.34. 6D does not require it, but suggests it as an optional standard.
608 has requirement for strength of ball to stem connection and location of weakest part of stem. 6D regulates strength of stem only.
608 requires adjustable packing glands. 6D does not.
For potential pressure increase from liquid trapped between the seats, 608 refers to ASME B16.34. 6D defines a performance test and acceptance limit.
608 defines allowable distruptions on the flange gasket face. 6D does not.
608 tests the valve per API-598, which suggests only low pressure tests recognizing the limitations of soft-seated floating ball valves. API-6D requires seat testing at 1.1 times rated pressure.
There is more so a side by side review of each is necessary to capture all the differences.