Painting with a knife is a bit like putting butter or jam on bread and produces quite a different result to a brush. Painting knives are excellent for producing textured, impasto work and sweeping areas of flat color as well as tiny shapes of color.
Although there is a difference between a painting knife and a palette knife, many people use the terms interchangeably. I don't see that it really matters. The main difference is, after all, that it's not a brush that you're using to paint with.
Strictly speaking, a palette knife is a long, straight blade or spatula that is used for mixing paints and scraping a palette clean, not for applying paint onto a canvas. A palette knife can be made from metal, plastic, or wood and will either be completely straight or have a slightly cranked (bent) handle.
A painting knife is most commonly made from metal with a wood handle, and has a large crank or bends in the handle, which takes your hand away from the painting surface and helps keep your knuckles out of the wet paint you've just applied. Painting knives come in numerous shapes (for example pear-, diamond-, or trowel-shaped) and are used for painting instead of a brush. There is, of course, nothing stopping you from using a painting knife for mixing paint on your palette.