To wear: to have something on your body
– She was wearing a diamond necklace with matching earrings.
– Who is that boy wearing a black jacket?
To put on: to move something you wear onto your body
– Take that shirt off and put on a new one. You can’t go out in such an old shirt.
– It’ll be more beautiful if I put on this skirt.
– It’s sunny today so I decided to put on my sunglasses.
– Today, I wear sunglasses because of the sun.