A practical recommender system should be able to handle heterogeneous behavioral feedback as inputs and has multi-task outputs ability. Although the heterogeneous one-class collaborative filtering (HOCCF) and multi-task learning (MTL) methods has been well studied, there is still a lack of targeted manner in their combined fields, i.e., Multi-behavior Multi-task Recommendation (MMR). To fill the gap, we propose a novel recommendation framework called Behavior-aware Variational AutoEncoder (BVAE), which meliorates the parameter sharing and loss minimization method with the VAE structure to address the MMR problem. Specifically, our BVAE includes address behavior-aware semi-encoders and decoders, and a target feature fusion network with a global feature filtering network, while using standard deviation to weigh loss. These modules generate the behavior-aware recommended item list via constructing better semantic feature vectors for users, i.e., from dual perspectives of behavioral preference and global interaction. In addition, we optimize our BVAE in terms of adaptability and robustness, i.e., it is concise and flexible to consume any amount of behaviors with different distributions. Extensive empirical studies on two real and widely used datasets confirm the validity of our design and show that our BVAE can outperform the state-of-the-art related baseline methods under multiple evaluation metrics.