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See Wikipedia on STL and COLLADA/DAE. Essentially, stl is the simpler variant of representing a mesh and only allows for surface geometry information. In contrast, dae files use XML and allow to embed much richer information, such as color, texture and other data. Both are supported by various ROS tools (rviz, moveit etc.) and Gazebo.

I'd recommend using stl if you don't need the additional capabilities dae offers, as it's simpler (and smaller in binary format).