Multi-tasking optimization can usually achieve better performance than traditional single-tasking optimization through knowledge transfer between tasks. However, current multi-tasking optimization algorithms have some deficiencies. For high similarity problems, the knowledge that can accelerate the convergence rate of tasks has not been fully taken advantages of. For low similarity problems, the probability of generating negative transfer is high, which may result in optimization performance degradation. In addition, some knowledge transfer methods proposed previously do not fully consider how to deal with the situation in which the population falls into local optimum. To solve these issues, a two-stage adaptive knowledge transfer evolutionary multi-tasking optimization algorithm based on population distribution, labeled as EMT-PD, is proposed. EMT-PD can accelerate and improve the convergence performance of tasks based on the knowledge extracted from the probability model that reflects the search trend of the whole population. At the first transfer stage, an adaptive weight is used to adjust the step size of individual's search, which can reduce the impact of negative transfer. At the second stage of knowledge transfer, the individual's search range is further adjusted dynamically, which can improve the diversity of population and be beneficial for jumping out of local optimum. Experimental results on multi-tasking multi-objective optimization test suites show that EMT-PD is superior to other six state-of-the-art evolutionary multi/single-tasking algorithms. To further investigate the effectiveness of EMT-PD on many-objective optimization problems, a multi-tasking many-objective test suite is also designed in this paper. The experimental results on the new test suite also demonstrate the competitiveness of EMT-PD.