Also, don't forget that the act of taking a measurement actually affects the circuit.
A DC Ammeter should have a low resistance, close to zero, but crucially NOT zero. Same, but opposite, for a DC Voltmeter; it should have high resistance, theoretically infinite, but it will just be very high.
This means that you never read the true value. Adding an ammeter will put a little more resistance in series, meaning a little less current flowing. Adding a voltmeter will add a very large resistance in parallel, meaning a little more current flowing, so more voltdrop in the series part of the circuit, than without it.
The effect will be dependant on the existing conditions of the circuit, in relation to the connection point, and value of resistance , of your test instrument. You can't take a measurement without affecting what you are measuring.
Biggest errors will be measuring current when you already have a low resistance circuit, or measuring voltage when you have a very high resistance circuit.
Using a resistor divider network and then measuring voltage at a midpoint will put the voltmeter in parallel with one of the resistors and mean the combined resistance value is now lower; 1/(1/R1+1/Rmeter). This will change your divider ratio and the voltage at that point to a lower volt drop than would be present without the meter connected. Therefore, the value with and without your meter connected WILL be different anyway, dependent on the values in the circuit and that of your meter.
Add on the errors mentioned previously and you will NEVER measure exactly the same with and without the meter connected, unless you actually had an infinite impedance voltmeters.
You could well be chasing ghosts here, by trying to get a value that is the same (or as close as you seem to want it), with all the factors that are affecting the measurement of that value via different methods.