We present optical and near-infrared spectroscopy (600-1000 nm) of eight faint (I > 18) very red (R - I > 2.2) objects discovered in a recent deep CCD survey of the Pleiades cluster by ZapateroOsorio, Rebolo, & Martin. We compare them with reliable cluster members like PP1 15 and Teide 1, and with several field very late type dwarfs (M4-M9.5), which were observed with similar instrumental configurations. Using pseudocontinuum ratios, we classify the new substellar candidates in a spectral sequence defined with reference to field stars of known spectral types. We also reclassify PP1 15 and Teide 1 in a self- consistent way. The likelihood of membership for the new candidates is assessed via the study of their photospheric features, Ha emission, radial velocity, and consistency of their spectral types and I-band magnitudes with known cluster members. Four of the new substellar candidates are as late or later than PP1 15 (M6.5), but only one, namely, Calar 3 (M8), clearly meets all our membership criteria. It is indeed an object very similar to the brown dwarf Teide 1. Out of the eight new substellar candidates, our "cautious" membership analysis leaves only Calar 3 as a Pleiades brown dwarf with a high level of confidence. This object, together with Teide 1, allows one to compare the spectroscopic characteristics of Pleiades brown dwarfs with those of old very cool dwarfs. The overall spectral properties are similar, but there are slight differences in the NaI doublet (818.3 nm, 819.5 nm), VO molecular band (740 nm), and some spectral ratios, which are probably related to lower surface gravity in the young Pleiades brown dwarfs than in field stars. Finally, we propose a way of improving future CCD-based brown dwarf surveys by using narrowband near-IR pseudocontinuum filters.