We present spatially resolved, near-diffraction-limited 10 μm spectra of the nucleus of the Seyfert 2 galaxy NGC 1068, obtained with Michelle, the mid-IR imager and spectrometer on the 8.1 m Gemini North Telescope. The spectra cover the nucleus and the central 60 × 04 of the ionization cones at a spatial resolution of approximately 04 (≈30 pc). The spectra extracted in 04 steps along the slit reveal striking variations in continuum slope, silicate feature profile and depth, and fine-structure line fluxes on subarcsecond scales, illustrating in unprecedented detail the complexity of the circumnuclear regions of NGC 1068 at mid-IR wavelengths. A comparison of photometry in various apertures reveals two distinct components: a compact (radius < 15 pc), bright source within the central 04 × 04 and extended, lower brightness emission. We identify the compact source with the AGN-obscuring torus, and the diffuse component with dust in the ionization cones. While the torus emission dominates the flux observed in the near-IR, the mid-IR flux measured with apertures larger than about 1'' is dominated instead by emission from the ionization cones; despite its higher brightness, the torus contributes <30% of the 11.6 μm flux in the central 12 region. Many previous attempts to determine the torus spectral energy distribution are thus likely to be significantly affected by contamination from the extended emission. The observed spectrum of the compact source is compared with clumpy torus models. The models require most of the clouds to be located within a few parsecs of the central engine, in agreement with recent mid-IR interferometric observations. We also present a UKIRT/CGS4 5 μm spectrum covering the R(0)-R(4) lines of the fundamental vibration-rotation band of ¹²CO. None of these lines was detected, and we discuss these nondetections in terms of the filling factor and composition of the nuclear clouds.