Yes, there's a difference. In the first case, you are interrupting the action defined by the gerund (-ing). In the second case, you are interrupting the action that is not defined in order to do the second action.
Case 1: Karen stopped asking for directions => In this case, she is in the process of doing the action specified by the gerund (asking).
She may be asking several people the same questiion and each person gives a different direction. So she decides to stop asking (not ask for directions). This is an interruption of an action.
Case 2: Karen stopped to ask for directions. In this case, she is doing something *else* that is unspecified, and this is what she stops doing in order to do the *second* action (ask), which is to ask for directions.
She may be driving a car, and this is what she stops doing in order to ask for directions. This is a change of actions.