There are more than a few cases that complicate the use of the words god and God.
The most obvious is that God is a proper noun used to name a particular god, and this leaves some possible difference as in:
Oh my God!
Oh my god!
Even if we were extremely strict to the use of capitals for God as a name, and assume the speaker is Christian (though if anything such exclamations should not be made by Christians in most contexts where they are heard), we could still consider either correct, the former addressing their god by name, the latter referring to Him by what He is.
Another issue is that god and God are used as a proper noun not just for the Christian god*, but for a variety of monotheistic concepts, including pantheism, deism, Neoplatonism and other concepts that combine monotheism and polytheism (consider the popularity in neopagan circles of Dion Fortune's forumla "All gods are one God, and all goddesses are one Goddess, and there is one Initiator"), and perhaps in particular in cases where one is deliberately quite vague and covering all such concepts. There are also translations from Eastern religion that use god or God with differing degrees with how well one could argue them relevant.
Now, here we may still use God with a captial as in the Fortune quote above (though Fortune was herself a Christian for all her unorthodoxy and would be inclined to mirror the conventions among them when not explicitly moving away from one for a reason). We might also though, use a non-captial form, as capitals are not always used with proper nouns (consider that when "history" is used as a proper noun as in "for all of history" it is normally not capitalised). This makes the form god also appropriate, especially the further we move from any precise view on which god is referred to, or in those cases in which the god believed in is not (or not necessarily) believed to be a person or persons in the same way that some religious views hold their god to be, making the capital form less applicable.
Now, if I say "god does not play dice" or "God does not play dice", which god do I refer to? Am I referring to the Judeo-Christian god because the man I am quoting was from a Jewish background, or am I being vauger since the phrase is after all figurative?
The answer in Western society is that it's somewhere between the two: We have a lot of different concepts of god and God both literal and figurative within our society, and we also have a strong influence of Christianity as the strongest religious influence in the English-speaking world for many centuries (with Judaism being the second strongest), but not the sole view; consider in particular the importance of Deism among Enlightenment thinkers whose influence on scientific, artistic and political thought persist to this day.
Some of this has been explicitly suggested too, mostly as justifications for religious intrusion into all manner of secular aspects of life with a claim that references to god do not necessarily refer to God.
Some of this is more implicit, when people of different beliefs - which may in themselves be quite vauge - speak in vague terms to each other.
So, it's quite sensible that we should end up with god and God both being used for the vaguely probably-Christian-but-perhaps-not-quite concept of divinity that is most often addressed.
And that's before we get to the fact that many people just simply won't care.
*Note that this is another point where I could validly use either, "the Christian god" meaning the god that Christians have and "the Christian God" meaning the god that the Christians have, who they refer to as God.