When making woodblock prints (also called woodcuts or wood engravings), the texture of the wood grain transfers better when using a hard brayer. Generally, a high-contrast impression is fundamental to woodblock printing, and a hard brayer allows the cut designs to show up well. It can be tricky, however, to distribute the ink evenly across the woodblock, especially if the surface is irregular or rough.
Soft brayers are more suitable for etching and other relief printing methods, as their flexibility makes it easier to cover the whole surface quickly and evenly. The finely-cut details on an etching plate transfer more effectively using a soft brayer. You can build up thin coats of ink with soft brayers, rather than putting on one thick layer.
Use a hard brayer when creating linoleum block prints (also called linocuts) for the same reason as woodcuts, although there is no wood grain to transfer. A distinct impression is essential to linoleum block prints. Using a hard brayer achieves this result. Linoleum blocks are very smooth, so it's easy to get uniform ink coverage across the surface.
Monotype printing is a one-print process that involves drawing or painting on a flat, smooth, non-porous surface. You can use a copper or zinc etching plate; you can also use glass or transparent acrylic plastic. The image is transferred simply by pressing the plate onto paper. Soft brayers are best for inking the monotype surface, because they offer you more control in applying pressure to create different effects.