There is certainly variation between the two, as they are calculated separately using different samples, but most times the difference is small. For example, in December, ADP showed an increase of 153,000 private sector payroll jobs, while BLS only showed an increase of 144,000.
So...you have claimed you'd believe private sector over government, but what do you do when they match?
For Unemployment, Gallup runs their own survey and while the differences between the Gallup and BLS rates can be large, they're never statistically significant in that BLS has a margin of error of +/- 0.2 percentage points at 90% confidence while Gallup has a an error of +/- 0.7 percentage points. So considering those to ranges always overlap and the BLS sample is about 4 times as large ...30,000 adults/month for Gallup and 60,000 households for BLS (close to 120,000 individuals)...and Gallup only counts age 18+ while BLS uses 16+ there's never any clear difference between the two.