A stain is any mark which an unwanted substance makes on a surface. If a cut finger drips blood on the carpet, for example, this drop will create a stain which is probably round or teardrop in shape . A cup of coffee might make a round stain on a table.

A smudge is a longer mark which occurs when the substance has pulled, rubbed or smeared across or along the surface. For example, you can smudge make-up if it is rubbed onto the wrong part of your face - if you rub your eyes, this might smudge your eyeliner.  A messy kiss from a child will smudge your lipstick. In the days when people used ink pens, you had to blot your writing so that the ink didn't smudge.  Smudges, by definition, are always thinner at one end.  Smudge and 'smear' are similar in meaning, and are both nouns and verbs.