From which activities am I (not) excluded by not registering for a conference/symposium? I'm a student about to start my master studies and I'm starting to have opportunities to take part in a symposium and a conference.

I have an accepted article at the conference and was therefore allowed to register for the conference. This involved paying (not on me) the registration fee etc.

I don't have an article for the symposium but the symposium interests me a lot. The chairs offered student support i.e. (partially) waive the registration fee for some students even if the student doesn't have an accepted article (although preference is given to those that do).

It turns out I wasn't granted this support. However, I'm still allowed to attend the talks without registering.

So, if I don't need to be registered to attend talks: what does it exactly mean to be registered for a symposium and/or conference? From which activities am I excluded by not registering and for which ones am I not excluded?

The symposium is abroad but I will be there around that time by coincidence. Is it worth staying there for a couple of extra days considering the activities I'm eligible for?
The terms "conference", "workshop", "symposium" might have specific meanings in some specific contexts, but in generall, they are rather interchangeable terms.

Registration fees are typically intended to cover the cost of the conference, and maybe to make some profit for the organization running it. In any case, part of the fee will cover the costs directly attributed to the person attending, and other parts go towards general costs.

Even if one is not causing any specific costs, it is often frowned upon to attend any part of a conference without being registered, unless this has been cleared with the organizers.

In your case, this happened, and the question is what exactly you might attend. The rule of thumb is that if there is a cost-component per delegate, stay away from it: Coffee breaks, lunch, conference dinner, conference excursion would be typical examples. Also, do not take a conference bag.

For most conferences, the rest will be talks (in various incarnations), and maybe poster sessions. If you find something else on the schedule, you should ask the conference organizers directly.