These two even have different aspects. A more meaningful question would be, how менять(ся) and изменять(ся) are different. Then поменять/разменять and so on naturally join the crowd :)
First, they are interchangeable to an extent, in SOME meanings. The basic difference is:
- менять = to "modify"(be modified) OR to "replace" something (be replaced by something else)
- изменять = to "modify" (be modified). And only that. The verb cannot mean that a thing not only changed its properties but got completely replaced by another thing.
Given that "менять" has a broader meaning, you can use different perfective counterparts for it:
- "change" as in "They changed their opinion". → изменить(ся), also поменять(ся).
- "change" in the meaning of "exchanging" things → обменять/поменять
- "change" as "Last year they changed the headmaster (replaced by a new one)" → заменить/сменить
- exchanging cash when your money is so big that a seller cannot give you change → разменять.
This is yet another evidence that on a more advanced level you find that the aspectual pairs are just a practical thing, not the truth central to the language. That is, you will always have an aspectual pair at hand when you need to convert an "event" into a "process" or "repeated event" (or vice versa). However, in reality each prefix has a cloud of meanings, and there are no prefixes that "just" convert a verb into a perfective.
The reason you have "pairs" is that for most verbs there is one prefix with a meaning that very much overlaps the verb's own meaning. For example, when you replace someone — then the "away", "off" meaning of the prefix "с-" overlaps your particular meaning of "менять": removing something and placing something else there. So "сменить" becomes a natural perfective counterpart. "Переменить" also works, to an extent, though I would not use it myself that often.
Also, "изменить"(изменять) has a second meaning: to betray, fail someone. In a more specific meaning, to cheat on your wife or husband (though still used in combinations like "My mind fails me").