Crayons are made primarily from wax, while oil sticks are made from a mixture of non-drying oils and waxes. There are many differences between the two besides the difference in composition:
When painting with crayons, it takes a bit more effort to paint a full area of color, but oil sticks are relatively easy and smooth for laying down large areas of color.
Oil pastels are very colorful and soft. So the colors are easy to mix and can be easily rubbed in with your fingers. Crayons, however, are harder, so the colors don't mix as well. It's very easy to get the color on your hand with a paint stick, but not so easy with a crayon.
Take the fruit for example, the first one is the brushstroke of the stick, you can see the thick creamy texture of the brushstroke, and you can also see the smoothness of the blue canvas rubbing over it.
Looking at the second picture for comparison, let's put aside the colors for a moment, because there is also a relationship between color saturation and the artist's own preferences. You guys look at the brush strokes, because the crayon is harder, it creates a relatively more delicate texture. It's similar to colored pencils, but with more of a wax sheen than colored pencils.
Oil sticks, because they are thicker, will have some of the layered buildup of an oil painting, whereas crayons may not do as well. A stick can cover a crayon image just as it can cover many other surfaces-glass, wood, canvas, metal, stone-but a crayon can only paint on paper.
But there's another way to play with crayons because they're made of wax: wax melts when it's hot, so you can take a hairdryer and play with the splatter effect. You can google melted crayon painting, and it will come up with a lot of things like this.
