Photoshop is primarily for raster images whereas Illustrator is for vector images..
both are very powerful programs.. I use both.. Photoshop for digital editing, sketching, and painting.. Illustrator for design and cartooning
I'll give an example of both something I've done in photoshop and illustrator that I think will outline the difference between producing images using raster graphical manipulation and vector graphical manipulation
Photoshop
Notice the individual strokes, subtle tonal variation, relatively high amounts of detail (although high detail can be acheived in illustrator it takes a higher degree of skill or time investment).. these are things that characterise the capabilities of photoshop
Illustrator
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Notice the bold lines, the cleanness and precision (acheived with none of those skills i might add), these are the kinds of things that are characterised by vector graphics in illustrator.
Both have their strengths and weaknesses and both have crossover (although fairly limited) between each other in terms of features and editing. Both our highly powerful and have massive time saving implications granted by the strengths that apply to them respectively