The short story is: one is content of something in particular, but heureux in general.
Je suis content de te voir. (I'm glad to see you.)
Je suis content que tu aies pu venir. (I'm glad you could come.)
Ils vécurent heureux et eurent beaucoup d'enfants. (They lived happily ever after — lit. They lived happ(il)y and had many children.)
Content is rarely used without a qualifier (possibly implied) indicating what the person is happy about, though you can find it occasionally in literary usage.
Heureux can be used to mean that one is happy about something in particular, but it has a requirement that content doesn't have, which is that the person who is described as happy should be involved in the event. This is a usage trend, not an absolute rule. For example, “(?) Je suis heureux que tu puisses venir” sounds awkward; on the other hand, “Je suis heureux d'apprendre que tu pourras venir” sounds fine (it's a fairly standard formal construction, where one states being happy not of the event but of learning about it).
Note that this is only the short story. There are many nuances, many cases where the two words can be used interchangeably or where only one of the words fits despite not fitting the general/particular mold.