I'm no expert on the physiology of stress myself. I've jus done a bit of reading on the subject.
You are both correct, there are differences between exercising strenuously for an hour a few times a week and constantly worrying about exams, bills, career etc.
The point is that there are also many similarities IE sympathetic nervous system tone/fight or flight etc.
The HUGE difference between the two situations is the frequency and duration of the stress. In essence, constantly freaking out about things doesn't allow ones body to 'calm down'. Meaning there is a constant amount of low intensity 'fight or flight' going on. It's not necessarily about the magnitude of the stress response at any given time point because them exercise would be horrible for you. It's more like an area under the curve type of deal. Exercise gives a lot of stress in a relatively short amount of time with a good amount of recovery time to allow for adaptation whereas finals leave one in a state of low level stress for very long periods of time.
You can also look up Selye's GAS model/principle for further reading.