Soft actuators that combine compactness, rapid response, and high output power are critical for advancing high-performance soft robotic systems. Although bistable architectures can amplify speed through elastic instability, their combination with functional materials often sacrifices compactness or lacks tunability. Here, we report a Venus flytrap-inspired shape memory alloy - embedded snapping actuator (SMA-ESA) that achieves compact and tunable actuation by structurally integrating pre-shaped SMA wires within a double-tilted elastomeric matrix. The SMA skeleton not only triggers snap-through via its thermal shape memory effect but also regulates performance. Combined experimental and finite element studies reveal that the actuation performance - characterized by energy storage capacity as well as the rate and efficiency of energy release - is tunable through both geometric parameters and input power. As a proof of concept, the SMA-ESA is demonstrated in a flytrap-inspired capture device that selectively responds to external stimuli. These results establish a generalizable strategy for embedding high-power-density materials into bistable soft structures, offering new opportunities for compact, responsive, and bio-inspired soft robotics.