Hi MHD. I also agree with Fraggle that contentment is more a state of being, while happiness is more often a response to a specific set of circumstances, and may be shorter-lived, or at least more variable in strength. I've had friends who've been depressed, and it has been very clear from talking to them that their feelings are very different from 'bog standard unhappiness', even compared to times in my life when I've felt very unhappy for a sustained period of time. I kind of wonder if contentment is, somehow, the reverse of that?
We also need to remember that 'to be content' also has the meaning of 'to put up with'. It may now be a slightly old-fashioned usage (I'm hearing in my head a slightly Jane Austen-ish sentence like 'Not having found a suitor to marry her, she contented herself with her embroidery' 
) but I think that the meaning is still somewhere contained in the word.
And it seems to me that this is what Paul intends in Philippians 4. I've just looked at the 'Blue Letter Bible' website, which allows you to compare different translations, and most have something like 'I am not saying this because I am in need, for I have learned to be content whatever the circumstances'; in other words, I can put up with anything [because of my faith]. Is that right?
The Vulgate translation here, interestingly, has 'enim didici in quibus sum sufficiens esse': roughly translated, for I have learnt to be satisfied in whatever state I am (sufficio - to be sufficient, to suffice, avail for, meet the need of, satisfy, according to Lewis and Short's Latin dictionary).
I don't read Greek, but Google is my friend , and it looks as if the Greek equivalent is αὐτάρκης [autarkes], which is defined, interestingly, as 'self-sufficient, independent, content'.
In relation to what I know about (medieval - so very feasibly not at all relevant to what you're interested in) discussions of Christian morality, this is an important idea, because the lack of contentment in this sense can lie at the basis of any/all of the Seven Deadly Sins. A lack of contentment with one's place in the world and a sense that one deserves more than one receives = Pride; a lack of contentment with what one has, in comparison to what others have = Envy; a lack of contentment caused by a perceived sense of wrong done to one by others = Wrath; a gloomy resignation caused by a lack of contentment with one's lot and a sense of not being able to do anything about it = Sloth; a lack of contentment with what one has, in terms of material possessions, leading to acquisitiveness = Avarice; a lack of contentment with merely sufficient food/drink leading to excessive indulgence = Gluttony; a lack of contentment within the limits of pre-marital chastity and faithfulness within marriage = Lust.
And the distinction between contentment and happiness is also relevant to ideas about Christian eschatology, where contentment (willing acceptance of our condition in the here-and-now) is fulfilled in the true happiness of eternity. This is implicit (and possibly even explicit, but I'd need to go back to the text and check) in Augustine's distinction between the City of God in Heaven (which is perfectly happy) and the City of God on pilgrimage in this life, which has perforce to cooperate with the City of Man and therefore is, at best, merely content.
Oops... I appear to have gone on a bit! 
Hope this is useful anyway.