We present the first systematic investigation of spectral properties of 17 Type Ic Supernovae (SNe Ic), 10 broad-lined SNe Ic (SNe Ic-bl) without observed Gamma-Ray Bursts (GRBs) and 11 SNe Ic-bl with GRBs (SN-GRBs) as a function of time in order to probe their explosion conditions and progenitors. Using a number of novel methods, we analyze a total of 407 spectra, which were drawn from published spectra of individual SNe as well as from the densely time-sampled spectra of Modjaz et al. (2014). In order to quantify the diversity of the SN spectra as a function of SN subtype, we construct average spectra of SNe Ic, SNe Ic-bl without GRBs and SNe Ic-bl with GRBs. We find that SN 1994I is not a typical SN Ic, in contrast to common belief, while the spectra of SN 1998bw/GRB 980425 are representative of mean spectra of SNe Ic-bl with GRBs. We measure the ejecta absorption and width velocities using a new method described here and find that SNe Ic-bl with GRBs, on average, have quantifiably higher absorption velocities, as well as broader line widths than SNe without observed GRBs. In addition, we search for correlations between SN-GRB spectral properties and the energies of their accompanying GRBs. Finally, we show that the absence of clear He lines in optical spectra of SNe Ic-bl, and in particular of SN-GRBs, is not due to them being too smeared out due to the high velocities present in the ejecta. This implies that the progenitor stars of SN-GRBs are probably He-free, in addition to being H-free, which puts strong constraints on the stellar evolutionary paths needed to produce such SN-GRB progenitors at the observed low metallicities.