In the real world, you draw with a pen or a pencil, and you paint with a brush and colours.
Paints are 'wet' - mainly water-based (water colours) or oil-based, or acrylic. If you paint, you dip a brush in the paints and apply the colour to the paper or canvas. The paint then dries.
I'm less sure about how you'd use the terms in digital art. Strictly speaking, you can't 'paint' using a computer. If you create a digital image, you could call it drawing, but adding colour to the image isn't really 'painting'.