DrDiettrich:
Voltages are not always constant, even if supplied by a battery. A DVM handles such fluctuations and noise in a different way than an ADC does. For increased precision the ADC sleep mode has to be used, so that fluctuations inside the processor will not affect the measured value. A 10 bit ADC also has a higher uncertainty (steps) than the ADC in a DVM. And the outside AREF may be slightly different from the internal AREF, used by the ADC.There are many factors that can affect measured values. Take the constants that give the values you trust most
Technically, what's in a DVM is still an ADC, just a different type. Most multimeters use an integrating ADC. The input signal is used to charge a capacitor for a fixed period of time, and then the capacitor is discharged down to 0 and the discharge is timed. Higher time = higher input signal. These tend to be pretty slow compared to other types of ADC, and the time it takes to convert a signal is different depending on its value.
The ATmega328P (and probably most other non-specialized microcontrollers) has a successive approximation converter. Its conversion procedure is a bit more complicated, but can work much faster and with a fixed conversion time.