Two issues surround the distinction between acts and omissions.
- How can we analyse this distinction?
- Is it morally relevant.
Utilitarians are committed to the position that the distinction is morally irrelevant. Act utilitarians, for example, will say that we should not act in a way that allows an upshot, denoted U, to occur in exactly the same way that we should not act in a way that causes U to occur.
Jonathan Bennett’s analysis of the distinction and argument against its moral relevance Edit
The answer to 1 above is clearly relevant 2. In The Act Itself, Jonathan Bennett defends an analysis of the distinction which he shows it to be clearly morally irrelevant. For descriptions of the analysis and argument, see Is the difference between making something happen and allowing it to happen morally relevant?