When your decision is between a plain colour and a pattern, your first thought should be paint for the plain colour and wallpaper for the pattern.
But that's only the first thought. A good scrubbable wallpaper in a solid colour may be a better choice in the kitchen than paint would. A stenciled "wallpaper" may be a better choice in a humid space like a bathroom, where the paper might be steamed off in short order.
A painted wall can also be much less costly than a papered wall. If you have something special in mind and the budget won't quite stretch to a wallpaper, a painted wall can be a better choice.
It can also be the only choice if you have something really special in mind. Portraits of your family? Your kid's favourite cartoon character? A mural depicting all the planets and stars you can see from your window? All become possible with a custom painted wall. But only if you're an artist.
Even with a painted, patterned wall, start with an honest reckoning of your talents. Try painting a bit of canvas or board and be brutally frank with yourself: will your wall look good?
If you're not going to create a stunning piece of artwork for your wall, you might be able to stencil a beautiful design. Stencilling involves a cut shape, and it requires only a steady hand, not a lot of artistic talent. Again, try it out first and see how it goes on a bit of board. Check not only how neatly you can reproduce the design, but how well you can space the repeats, how evenly they butt up against one another, and how straight you can keep the rows.
If you've got the knack, think about the amount of time involved. A stenciled pattern on a wall may require you to paint the same shape hundreds of times, with a great deal of concentration required for even rows and a balanced pattern. You have to do the maths involved in calculating how to get the pattern properly centered on the walls and make it fit correctly at the corners.
Wallpaper can be a lot less trouble.
If you still want a painted pattern on your wall, or a mural, and you've decided that doing it yourself isn't realistic, contact a
professional painter. Better yet, a firm of painters and decorators who do both wallpaper and paint. They'll be best able to tell you whether paint or paper will be the perfect choice for what you have in your mind's eye.