> Is there a distinction between government and religion in Islam?

No more or less than there is in Christianity.

There are Muslims who distinguish the role of each, just as there are Christians who do.

> In the west, there is this distinction

Not largely from within institutional Christianity, where both theocracy (wherein government is viewed as subordinated to religious institutions, rather than the two being separate) and caesaropapism (wherein religious institutions are subordinated to government rather than the two being separate) have been common institutional power and legitimate government power is frequently portrayed as having a divine religious origin.