Diffractive neural networks (NNs) have garnered attention for directly implementing wireless signal processing at the physical layer. However, they are limited by a constrained weight learning space and activation functions, which restricts their data processing capabilities. To address this, we propose an RF circuit-based weighted diffraction NN (WDNN) that rivals digital NNs in processing ability. We design a weighted asymmetric RF coupler unit that, when stacked into a network, enables diffractive propagation with arbitrary connection weights. Additionally, an activation module is introduced that utilizes RF amplifiers operating in their nonlinear regions. We validate the effectiveness of the proposed WDNN through three tasks: 32-level amplitude modulated (AM) signal decoding, 31-class angle of arrival (AoA) estimation, and 2-class Wi-Fi based fall detection. After training, WDNN achieves the accuracy of 98.5%, 93.7%, and 90.8% in the AM decoding, AoA estimation, and fall detection tasks, respectively; while the diffractive NN SOTA achieves only 21.6%, 16.9%, and 63.3%. We also implement the prototypes of WDNN and SOTA, and real-world experimental results demonstrate that our method achieves an average accuracy improvement of up to 76.85% across various tasks compared to SOTA.