No practical body fat method measures body fat. Each one measures something easier, then predicts fat percentage from it using an equation fitted to a reference population.
Circumference methods measure girths and predict from those. Skinfold methods pinch subcutaneous fat at set sites and predict from the sum. Both are reasonable. Neither is looking at fat directly, and they will not return the same number.
Two methods on comparable inputs
A circumference estimate for a 175 centimetre, 75 kilogram man with a 38 centimetre neck and an 88 centimetre waist comes out at 19.2 per cent, which splits the weight into 14.4 kilograms of fat and 60.6 kilograms of lean mass.
A three site skinfold estimate summing 41 millimetres returns 12.4 per cent by way of an estimated body density of 1.070 grams per millilitre. Different route, different assumptions, different answer.
Where each one goes wrong
Circumference formulas assume fat is distributed in a typical pattern for your sex and build. Someone who carries fat unusually, or has a thick neck from training, breaks that assumption in a way the formula cannot detect.
Skinfolds are more direct about the tissue in question but depend completely on technique. Site location, pinch depth, caliper pressure and timing all move the reading, and self-measurement is notoriously inconsistent because two of the three sites are hard to reach reliably on yourself.
The number that matters is the change
Because each method carries a systematic bias, the absolute figure is the least reliable thing it produces. The useful output is the direction and size of change, measured the same way, by the same person, under the same conditions.
Switching methods mid-programme throws that away. A drop from 19 per cent to 12 per cent that coincides with switching from a tape measure to calipers is a change of instrument, not of body.
What to hold the figure against
Body fat percentage is one input among several, and it is at its worst when treated as a target with a deadline. Strength, how clothes fit, energy and blood markers all describe things a single percentage cannot, and none of them require an equation.
The body fat calculator uses the circumference method, and the skinfold calculator uses the three site approach with the caveat about technique stated on the page. For anything clinical, laboratory methods exist and a professional is the right route to them.