Interactions between the cardiac and respiratory systems play a pivotal role in physiological functioning. Nonetheless, the intricacies of cardio-respiratory couplings, such as cardio-respiratory phase synchronization (CRPS) and cardio-respiratory coordination (CRC), remain elusive, and an automated algorithm for CRC detection is lacking. This paper introduces an automated CRC detection algorithm, which allowed us to conduct a comprehensive comparison of CRPS and CRC during sleep for the first time using an extensive database. We found that CRPS is more sensitive to sleep-stage transitions, and intriguingly, there is a negative correlation between the degree of CRPS and CRC when fluctuations in breathing frequency are high. This comparative analysis holds promise in assisting researchers in gaining deeper insights into the mechanics of and distinctions between these two physiological phenomena. Additionally, the automated algorithms we devised have the potential to offer valuable insights into the clinical applications of CRC and CRPS.