She rubbed (at) the stain on her shirt.
A similar case:
He wiped (at) the stain with the face towel.
Is the meaning the same? Or different?
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You can rub your cat to please it, or rub a lamp to make the genii appear.
When you rub at a stain it's to make it go away.
Since wiping is usually meant to make something go away, there is no need for the "at". You might use it if the wiping were unsuccessful.
Rubbing is more vigorous than wiping. You rub at a stain, but wipe up a spill. Wiping a stain often doesn't work, so you might say "wipe at" then, with a hint that the wiper was wasting time.
“At” can be another word for “toward”, implying a target or a goal, e.g., running at something.
The target or goal is usually implied by the context of the statement - in this case we would most likely understand it to mean that the rubbing was intended to remove the stain.
Another example of this kind of phrasing would be:
He kicked (at) the door.
You’ll also notice that this kind of phrasing sort of implies more than one occurrence of the activity, more than one rub or kick, and makes it feel like more of an ongoing activity, as opposed to a single action - again, like working towards a goal.