We present our Spitzer IRS spectroscopic survey from 10µm to 37µm of the Seyfert galaxies of the 12µm Galaxy Sample, collected in high resolution mode (R∼ 600). The new spectra of 61 galaxies, together with the data we already published, gives us a total of 91 12µm Seyfert galaxies observed, out of 112. We discuss the mid-IR emission lines and features of the Seyfert galaxies, using an improved AGN classification scheme: instead of adopting the usual classes of Seyfert 1's and Seyfert 2's, we use the spectropolarimetric data from the literature to divide the objects into categories "AGN 1" and "AGN 2", where AGN 1's include all broad-line objects, including the Seyfert 2's showing hidden broad lines in polarized light. The remaining category, AGN 2's contains only Seyferts with no detectable broad lines in either direct or polarized spectroscopy. We present various mid-IR observables, such as ionization-sensitive and density-sensitive line ratios, the PAH 11.25µm feature and the H 2 S(1) rotational line equivalent widths, the (60µm -25µm) spectral index and the source extendedness at 19µm, to characterize similarities and differences in the AGN populations, in terms of AGN dominance versus star formation dominance.We find that the mid-IR emission properties characterize all the AGN 1's objects as a single family, with strongly AGN-dominated spectra. In contrast, the AGN 2's can be divided in two groups, the first one with properties similar to the AGN 1's except without detected broad lines, and the second with properties similar to the non-Seyfert galaxies, such as LINERs or starburst galaxies.We computed a semianalytical model to estimate the AGN and the starburst contributions to the mid-IR galaxy emission at 19µm. For 59 galaxies with appropriate data, we can separate the 19µm emission into AGN and starburst components using the measured mid-IR spectral features. We use these to quantify the brightness thresholds that an AGN must meet to satisfy our classifications: AGN 1 have an AGN contribution ≥ 73 % and AGN 2 ≥ 45 % of their total emission at 19µm.The detection of [NeV] lines turns out to be an almost perfect signature of energy production by an AGN. Only 4 (∼ 7.5% percent) of 55 AGN 1 and 2 (10% percent) out of 20 AGN 2 do not have [NeV] 14.3µm down to a flux limit of ∼ 4 × 10 −15 ergs −1 cm −2 . We present mean spectra of the various AGN categories. Passing from AGN-dominated to starburst-dominated objects, the continuum steepens, especially at wavelengths shorter than 20µm, while the PAH feature increases in its equivalent width and the high ionization lines decrease.We estimate H 2 mass and excitation temperature through the measurement of the S(1) rotational line of this molecule. Finally we derive the first local luminosity functions for the brighest mid-infrared lines and the PAH feature at 11.25µm. No statistical difference is apparent in the space densities for Seyfert 1's and 2's of a given line luminosity, nor for the new classes of AGN 1's and 2's. We use the correlation between [Ne V] line ...