Both pressure and stress are defined in terms of force per unit area. Beyond that simple relationship, what might constitute a good intuitive way to distinguish these two concepts?
They are basically the same thing and the terms used tend to reflect context rather than physics. You would use 'Stress' when dealing with stretching etc, pressure when dealing with gases etc. Similar 'problems' occur with Newton.metres which can mean work or energy (Joules) or moment as in turning effect and torque
Nm is strictly a unit of Moment. The 'Newtons times metres' that is used to calculate Work is the Scalar Product of two Vectors (Force and Displacement) and Nm should never be used to describe Work of Energy. Pressure and stress, however, are totally equivalent and it's only the context that determines which term you want to apply to a situation. (It's not really a "problem")
Stress is a little more general than pressure- one intuitive way to think about stress is to think about how a square can be deformed. The square can be made larger or smaller (uniform expansion/compression), but also be sheared into a rhombus shape (and combinations of expansion with shear as well). 'stress' is the most general way to describe any deformation, while pressure is restricted to the uniform expansion/compression. Mathematically, pressure is the isotropic component of the stress tensor.