This paper considers a challenging scenario of machine type communications, where we assume internet of things (IoT) devices send short packets sporadically to an access point (AP) and the devices are not synchronized in the packet level. High transmission efficiency and low latency are concerned. Motivated by the great potential of multiple-input multiple-output non-orthogonal multiple access (MIMO-NOMA) in massive access, we design a grant-free MIMO-NOMA scheme, and in particular differential modulation is used so that expensive channel estimation at the receiver (AP) can be bypassed. The receiver at AP needs to carry out active device detection and multi-device data detection. The active user detection is formulated as the estimation of the common support of sparse signals, and a message passing based sparse Bayesian learning (SBL) algorithm is designed to solve the problem. Due to the use of differential modulation, we investigate the problem of non-coherent multi-device data detection, and develop a message passing based Bayesian data detector, where the constraint of differential modulation is exploited to drastically improve the detection performance, compared to the conventional non-coherent detection scheme. Simulation results demonstrate the effectiveness of the proposed active device detector and non-coherent multi-device data detector.