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The current study is concerned with the dynamical investigation, synchronization, and engineering application of a new variable-order fractional neural network. The model of the variable-order fractional neural network is presented, and its chaotic behavior is studied through well-known dynamical tools. Then, a new control technique is proposed for the control of the system. Although finite-time estimators considerably enhance the performance of controllers, studies that offer finite-time estimators for the control of fractional-order systems are rare in the literature. Motivated by this, as a novel approach, the proposed control technique is equipped with a finite-time estimator, which is able to approximate highly nonlinear disturbances and uncertainties. The stability and finite-time convergence of the sliding surface and error dynamics based on the proposed control technique are proven. Through numerical simulations, the effectiveness of the designed control scheme in the presence of complex time-varying disturbances is illustrated. Then, a voice encryption application has been implemented in order to show the feasibility of data security applications of the proposed variable-order neural network. Finally, to investigate the effectiveness of the implemented data security application, the entropy values for original, encrypted, and decrypted voice data are presented. Numerical analyses of encryption clearly confirm that encryption is done securely and there is no corruption or data loss in encryption and decryption processes. [ABSTRACT FROM AUTHOR]