One place where the distinction can become vital is in ER rooms. The following case was described (in another, closed, forum) by the chief on an actual ER: Suppose your ER is full, but you get a call about a patent in critical condition who needs an ER slot, and your ER is nearest. You have one slot occupied by a patient in a relatively stable condition, who could, with low risk, be moved to another ER, clearing the slot for the more critical case.
The utilitarian response is to accept the new patient, as the certainty of helping them outweighs the low risk in moving the existing one.
The deontological response is to refuse; you have accepted duty of care to the current patient, but not yet to the new one.
I don't know what a Virtue ethicist would do.
Actual ERs in the UK have different policies, depending on whether their ethics policy was written by a utilitarian or a deontologist. My interlocutor said that he would usually respond by the deontological rules, but in a crisis would (and in fact, had) act in a utilitarian way.
A deontological person can still practice triage. Without more information (such as published policies, prior agreements, patient consent, etc.) it is hard to say if the question is even really a moral dilemma. If you are standing on the street and two people get hit by a bus, deciding which one to help is not a question of morality. By helping either, or both, you are acting morally, and perhaps even supererogatorily. Deciding whether or not you should go kick them while they're down, though, is a moral question. It would be moral (though not particularly praiseworthy) to refrain from doing so, and it would be clearly immoral to kick them.
Interestingly, Catholics actually believe in the idea of a moral safe harbor. That is, if you've given serious and reasonable consideration to a moral question, and act in accordance with your earnest conclusion, then you will be held blameless regardless of the ultimate righteousness of the action.