We investigate spatio-temporal patterns occurring in a two-layer multiplex network of oscillatory FitzHugh-Nagumo neurons, where each layer is represented by a nonlocally coupled ring. We show that weak multiplexing, i.e., when the coupling between the layers is smaller than that within the layers, can have a significant impact on the dynamics of the neural network. We develop control strategies based on weak multiplexing and demonstrate how the desired state in one layer can be achieved without manipulating its parameters, but only by adjusting the other layer. We find that for coupling range mismatch weak multiplexing leads to the appearance of chimera states with different shapes of the mean velocity profile for parameter ranges where they do not exist in isolation. Moreover, we show that introducing a coupling strength mismatch between the layers can suppress chimera states with one incoherent domain (one-headed chimeras) and induce various other regimes such as in-phase synchronization or two-headed chimeras. Interestingly, small intra-layer coupling strength mismatch allows to achieve solitary states throughout the whole network.In nonlinear dynamics the paradigmatic FitzHugh-Nagumo (FHN) system is used to model the behavior of neurons. A single-layer network of nonlocally coupled oscillatory FHN neurons demonstrates various dynamic regimes including chimera states, that represent an intriguing mechanism of transition from complete coherence to complete incoherence. Here, we focus on a two-layer multiplex network and develop the tools for controlling the spatio-temporal patterns in the presence of weak coupling between the layers, i.e. weak multiplexing. We show that multiplexing combined with a mismatch between the layers, allows to induce chimera states in networks, where they do not appear in isolation. Our control also works in the opposite direction leading to the suppression of chimera states. Interestingly, small mismatch in the intra-layer coupling strength results in the formation of solitary states, that offer, compared to chimera states, an alternative scenario of transition from coherence to incoherence. Since multilayer structures naturally occur in neural networks (e.g., brain networks), we expect that our results can be useful for understanding and controlling of biological networks.