Utilizing evolutionary algorithms (EAs) to solve multitask optimization problems (MTOPs) simultaneously has gained significant attention because many real applications usually share similar properties or have a certain degree of correlations. In this new research area in the EA community, namely multitask optimization (MTO), a key challenge is transfering useful knowledge among different tasks efficiently. In this paper, a multitask differential evolution (DE) algorithm is proposed in which a simple variant of the canonical DE is adopted as the basic optimizer. Furthermore, an expert library module (ELM), which can extract useful knowledge from the historical optimization experiences of DE, is introduced as a complement for the DE to generate offspring. In the proposed algorithm, named MTDE-ELM, when the DE performs the mutation operator, if all individuals chosen to perform the mutation have the same skill factor, intratask knowledge transfer can be realized. Conversely, when individuals with different skill factors are adopted to execute the mutation, intertask knowledge transfer can be realized. In the ELM, evolutionary data of different tasks are used to train the feedforward neural networks (FNNs). After that, the trained FNN can realize potential knowledge transfer among different tasks. Moreover, based on the trained FNN, each individual can query its promising evolutionary direction, which can be regarded as an effective supplement for the DE to generate offspring. The comprehensive performance of the MTDE-ELM is demonstrated through comparisons with five other MTO algorithms. Furthermore, distinct properties of the newly introduced strategies are also confirmed by experimental analysis.