Short and generalized answer—The paths are the same, the output is different.
There are great explanations of the difference between vector and raster images here:
Slightly longer answer—It depends...
Adobe Illustrator is a vector graphics program. Everything you create in Illustrator is a vector, all of your colors and strokes and shapes are vector (although you can import raster images and use raster effects).
Adobe Photoshop is a raster image editor. It works exclusively with pixels. Paths in Photoshop are vectors, but what is output and what you see is a raster image based on that vector data. They are only vectors inside Photoshop, as soon as you export your image they are no longer vector. You can export an EPS from Photoshop which will retain some of your vector paths, but any fills in those paths will be raster images.
You can save as a Photoshop PDF too (not a regular PDF), which will retain some vector data, but other things will be rasterized (I believe any faux type styles are rasterized, for example).
If you want vector images, create them in Illustrator, not Photoshop.
But, if you already have artwork in Photoshop that was created with paths and shapes, not drawn pixels, you can resize your artwork in Photoshop with no problem.