The organelle responsible for packaging and distributing proteins in the cell is the Golgi apparatus. This structure modifies, sorts, and packages proteins produced in the endoplasmic reticulum for delivery to their destined locations, either inside or outside the cell. The Golgi apparatus plays a critical role in processing proteins with added functional groups, ensuring they are correctly folded and functional. It is essential for maintaining cellular function and communication, influencing processes such as secretion and cell signaling.