Stress is pain in motion. Actual pain, is consistent in the moment. Everyone loves to rave about pain endings etc, being the cause. But you can have pain regardless of situation, in the form of emotion. Emotions and stress are far more similar.
Pain and stress are not the same. While one can cause the other and vice versa, they are f.e. perceived in (mostly) different brain regions. The perseption of pain is altered by ones stress level.
Pain, in forms of physical pain like cutting your finger, is perceived via pain receptors through our spinal cord to the brain. But there is a "psychological" pain too, witout nerval stimuli of the body. There is some evidence (source: Empathy for social exclusion involves the sensory-discriminative component of pain: a within-subject fMRI study), that social exclusion triggers the same/ similar pain response systems as physical pain.
I would consider stress as a normal reaction from our body. There are stress hormones (like cortisol or adrenalin), which are released in a stressfull event. Of course feeling pain is stressfull. But we can feel pain regardless of whether we are stressed or not. Stress is a normal physiological response to make us ready to adapt to change. A chronical amount of stress is considered pathological. Pain is always a warning signal (to indicate that we are hurt - physically or psychologically).