Yes, ADP is more accurate with private sector jobs and BLS is more accurate with government jobs, because their data sources are heavily biased toward having accurate data in those subsets respectively. They both generally assume that the data they don’t have looks like the data they do have despite the selection bias and disjointedness of their data. BLS extrapolates private sector employment from government employment and census data in ways that no one thinks is representative but that is what they work with. ADP has no government employment data in their model but has a relatively exhaustive sample of private sector employment.
People that know account for these biases. This is why the ADP > BLS heuristic exists. If private sector jobs growth is less than government job growth, reflecting the biases in their respective models, that is interpreted as either the government sandbagging the numbers with make-work jobs or private sector employment being so poor that the monotonic government employment eclipses private sector job growth. Either way, it is a bad sign.
People that know account for these biases. This is why the ADP > BLS heuristic exists. If private sector jobs growth is less than government job growth, reflecting the biases in their respective models, that is interpreted as either the government sandbagging the numbers with make-work jobs or private sector employment being so poor that the monotonic government employment eclipses private sector job growth. Either way, it is a bad sign.