Is there a distinction between Morality and Justice? If so, what is it?

Usually justice is treated as a subset of morality. It's the part of morality that's concerned with:
1. distributive justice, e.g. the idea that the gap between the haves and the have nots should be bridged
2. property e.g. I should be able to keep what I have earned
3. reciprocity and contracts e.g. I did chores yesterday, it is only fair that you should do them today.

As you will see, the conventional notion of justice doesn't align perfectly within utilitarianism, but for the most part, it can be retained.

1. It is easier to improve the lives of the poor than to improve the lives of the rich. So utilitarianism will incidentally close the gap between the rich and poor.

2. We should respect each others property so that there is an incentive to work

3. We depend on each other's promises and social contracts in order to predict one another's behaviour. We must fulfil our promises except in severe extenuating circumstances so that they will continue to be made.

Justice and happiness tend to coincide, and justice is something we can grasp pretty intuitively. So when there's little time to decide the right course of action, taking the just course is usually good enough. I hope that answers your question and a few other ones along the way