The quite simple answer is that YES you are already treated differently to any other patient if you have covid. It's a highly infectious disease that needs certain protocols to prevent spread, vs say a heart attack patient. Will you be given lessor standard of treatment if you are vaxxed or un-vaxxed, NO, that will never happen as it would be against ethics, morals and basic human rights. The doctors may not even know who is vaccinated or not at presentation, it may be a question that does alter your treatment as presence of vaccine could alter the mix of drugs they give you to assist your recovery, so there may be some differences in treatment. Could you be triaged out of treatment if hospital capacity is exceeded, this is possible, as the capacity for a hospital to handle covid specific cases could mean you have to travel further, or are simply sent home to recover if not severe enough. But I would say there would be NO preference for Vaxxed vs not in any triage event, it would be on severity vs what help could be administered.
The main fact is that the more people that are vaccinated, the less that will present to hospital with severe symptoms and the less chance of the system reaching a point that turning patients away happens.