Define projection vs. project tool

The define projection tool overwrites the coordinate system information (metadata) stored with a dataset. If the coordinate system for a dataset is unknown or incorrect, you can use this tool to specify the correct coordinate system. You must know the correct coordinate system of the dataset before using this tool.  This tool only updates the existing coordinate system information; it does NOT modify any geometry.

The project tool transforms the geometry of the spatial data from one coordinate system to another.