A voltage is by definition a potential difference between 2 electrodes.
Nonetheless, to record a signal on 1 electrode that is hardly affected by the other reference electrode a distant electrode can be used, referred to as monopolar recording. By contrast, when 2 electrodes are placed close together on the site of interest it is referred to as a bipolar recording. In EEGs, it has everything to do with differential amplifiers and common mode rejection, which you already hint at in the question.
Without going into the technical details of opamps, I can say that differential amplifiers, ideally, reject the common signal on the electrodes and keep the differential voltage. In effect, most of the artifacts are rejected, while the signal is maintained. Artifacts are often widespread across the scalp, such as muscular artifacts, and are shared across electrodes, while the signal of interest is mostly confined to local areas on the scalp.
In EEG monopolar recordings are typically used, basically because sufficient artifact is rejected yet the signal is least distorted. In addition, and not least important, standard EEG electrode montages allow for the arithmetic deduction of bipolar recordings from monopolar recordings, whereas the reverse is not possible (Saab, 2009).
However, many electrophysiological signals, even those closely related to EEG, are often recorded using bipolar recordings, because either the signal is too low to be recordable, or the artifact is otherwise too large (e.g., Diez et al, 2010; Stronks et al., 2009; Stronks et al., 2013).
- Diez et al., Annu Int Conf IEEE Eng Med Biol Soc (2010): 5803-6
- Saab, Biofeedback (2009); 36(4): 128–33
- Stronks et al.,IOVS (2013); 54:3891–901
- Stronks et al., Doc Ophthalmol (2016); 132:1–15