We used the Atacama Large Millimeter/Submillimeter Array to map the CO(3-2) and the underlying continuum emissions around the type-1low-luminosity active galactic nucleus (LLAGN; bolometric luminosity less than or similar to 10(42) erg. s(-1)) of NGC 1097 at similar to 10 pc resolution. These observations revealed a detailed cold gas distribution within a similar to 100 pc of this LLAGN. In contrast to the luminous Seyfert galaxy NGC 1068, where a similar to 7 pc cold molecular torus was recently revealed, a distinctively dense and compact torus is missing in our CO(3-2) integrated intensity map of NGC 1097. Based on the CO(3-2) flux, the gas mass of the torus of NGC 1097 would be a factor of greater than or similar to 2-3 less than that found for NGC 1068 by using the same CO-to-H-2 conversion factor, which implies less active nuclear star formation and/or inflows in NGC 1097. Our dynamical modeling of the CO(3-2) velocity field implies that the cold molecular gas is concentrated in a thin layer as compared to the hot gas traced by the 2.12 mu m H-2 emission in and around the torus. Furthermore, we suggest that NGC 1097 hosts a geometrically thinner torus than NGC 1068. Although the physical origin of the torus thickness remains unclear, our observations support a theoretical prediction that geometrically thick tori with high opacity will become deficient as AGNs evolve from luminous Seyferts to LLAGNs.