The major contrast here is between what are known as deontological and consequentialist ways of thinking about ethics. In other words, the main difference between Kantian and utilitarian ethics is that under the Kantian system, one must do one's duty regardless of outcomes or consequences, whereas for the utilitarians, the consequences are the standard by which the ethical values of actions are judged.
The notion of moral rights is not essentially compatible with utilitarianism, which is a fundamentally hedonistic system of ethics. For the utilitarian, the ideal is happiness or well-being, and thus a good action is one leading to the greatest amount of well-being for the greatest number of sentient beings. Even Mill's harm principle does not entail the notion of a moral right.
For Kant, on the other hand, there are well defined moral rights, encapsulated in his imperatives. The most important is that every intelligent being must be treated as an end in itself rather than as a means to an end.