I am thinking about petty personal disputes, say if one person finds something that another person does annoying. A common gut reaction is to immediately start staking territory about what is just and what is virtuous and so on, while the correct thing to do is focus on concrete benefits and costs of actions. The main reason this is better is not because it maximizes utility but because it minimizes argumentativeness.

Another good example is competition for a resource. Sometimes one feels like one deserves a fair share and this is very important, but if you have no special need for it, nor are there significant diminishing marginal returns, then it's really not that big of a deal.

In general, intuitive deontological tendencies can be jerks sometimes, and utilitarianism fights that.

http://lesswrong.com/lw/b4f/sotw_check_consequentialism/