Every robotic gripper requires an equilibrated solution towards the grasp adaptability, precision, and load bearing capacity. A versatile soft robotic gripper requires adjustable grasp mode, for objects with different sizes and shapes, and adjustable compliance, for switching between soft mode; for small loads and delicate objects; and stiff mode; for larger loads and heavier objects. In this paper, we present the design of a tendon-driven robotic origami, robogami, gripper that provides self-adaptability and inherent softness through its redundant and under-actuated degrees of freedom (DoF). Robogami is a planar and foldable robotic platform that is scalable and customizable thanks to its unique layer-by-layer manufacturing process. The nominally 2D fabrication process allows embedding different functional layers with a high fidelity. In particular, a polymer layer with adjustable stiffness enables the independent control of the stiffness for each joint. Using this feature, we can control the input energy distribution between different joints and hence the motion of the robogami. Here, we model the behavior of a single finger; and demonstrate the compliance control of the end-effector along different directions in simulations and experiments. We also validate the gripper's task versatility in soft and stiff modes by assigning model-based joints stiffness for performing different grasp modes.