For the 6G wireless networks, achieving high-performance integrated localization and communication (ILAC) is critical to unlock the full potential of wireless networks. To simultaneously enhance localization and communication performance cost-effectively, this paper proposes sparse multiple-input multiple-output (MIMO) based ILAC with nested and co-prime sparse arrays deployed at the base station. Sparse MIMO relaxes the traditional half-wavelength antenna spacing constraint to enlarge the antenna aperture, thus enhancing localization degrees of freedom and providing finer spatial resolution. However, it also leads to undesired grating lobes, which may cause severe inter-user interference for communication and angular ambiguity for localization. While the latter issue can be effectively addressed by the virtual array technology, by forming sum or difference co-arrays via signal (conjugate) correlation among array elements, it is unclear whether the similar virtual array technology also benefits wireless communications for ILAC systems. In this paper, we first reveal that the answer to the above question is negative, by showing that forming virtual arrays for wireless communication will cause destruction of phase information, degradation of signal-to-noise ratio and aggravation of multi-user interference. Therefore, we propose the hybrid processing for sparse MIMO based ILAC, i.e., physical array based communication while virtual array based localization. To this end, we characterize the beam pattern of sparse arrays by three metrics, demonstrating that despite of the introduction of grating lobes, sparse arrays can also bring benefits to communications thanks to its narrower main lobe beam width than the conventional compact arrays. Extensive simulation results are presented to demonstrate the performance gains of sparse MIMO based ILAC over that based on the conventional compact MIMO.