Difference between saving as a Path vs Clipping Path

Wikipedia gives this definition for clipping path: "A clipping path is a closed vector path, or shape, used to cut out a 2D image in image editing software."

Years ago, we dealt with issues such as no transparency in tiffs or psds, pngs did not yet exist, and psds could not be placed into desktop publishing applications. The way to simulate transparency was to draw a path in PS on tiffs and jpegs, name the path, then place the image into the DTP app. 

This workflow continues today because a clipping path is a non-destructive way to simulate transparency. In InDesign you can apply a path that was created and named in Photoshop with this command: "Object > Clipping Path". (Alpha Channels are another choice, btw.) By default, Text Wrap automatically uses whichever path (or alpha channel) is set in that dialogue.

https://helpx.adobe.com/nz/indesign/user-guide.html/nz/indesign/using/clipping-paths.ug.html

~ Jane

And a curious detail about Clipping Paths (as used in eps): The Flattness-setting would determine how the Path would be interpreted on a postscript-output device (like a platesetter). 

Way back Clipping Paths were for eps – a pixel image that can be clipped by a Path when placed in a layout application. 

I think they are currently also used by Indesign when placing a tif, but Indesign can use non-Clipping Paths as well. 

When you clip a layer to a layer the clipped layer is visible  only where the its lower layer it is clopped to would be visible.  Basically the layer  is masked  to the lower layer transparency.  The lower layer could have a vector mask, a layer mask; or just have a smaller them canvas size layer bounds. The layer you clip can also be masked.