They differ first in their conception of what that happiness consists in...
They differ first in their conception of what that happiness consists in, secondly in views of how an agent's own personal happiness is aligned with, or traded against, the general happiness, and thirdly in whether it is necessary to acknowledge any other end for human action.
The simplest doctrine is that happiness is itself quite straightforward, consisting for example in occasions of pleasure; that agents only do seek or ought to seek their own happiness; and that there is no other possible or desirable end of action. The Cyrenaica may have held a doctrine along these lines. Complexity arises with more subtle conceptions of the nature of happiness.
Finally, theories of ethics that are not consequentialist in nature may recognize other ethically important features of action than those arising from the goal of maximizing either personal or social happiness (Blackburn, 2005). In ordinary use, the word 'happiness' has to do with one's situation (one is fortunate) or with one's state of mind (one is glad, cheerful) or, typically, with both. These two elements appear in different proportions on different occasions.
If one is concerned with a long stretch of time (as in 'a happy life'), one is likely to focus more on situation than on state of mind. In a short period of time, it is not uncommon to focus on states of mind. By and large philosophers are more interested in longterm cases. One's life is happy if one is content that life has brought one much of what one regards as important.
There is a pull in these lifetime assessments towards a person's objective situation and away from the person's subjective responses. The important notion for ethics is 'wellbeing' -that is, a notion of what makes an individual life go well. 'Happiness' is important because many philosophers have thought that happiness is the only thing that contributes to wellbeing, or because they have used 'happiness' to mean the same as 'wellbeing'. What, then, makes a life go well?
Some have thought that it was the presence of a positive feeling tone. Others have thought that it was having one's desires fulfilled - either actual desires (as some would say) or informed desires (as others would say). It is unclear how stringent the requirement of 'informed' must be; if it is fairly stringent it can, in effect, require abandoning desire explanations and adopting instead an explanation in terms of a list of good-making features in human life (Griffin, 2000).