Any Christian fundamentalist who advocates for its religion to become law is a bad Christian who never understood the lesson behind "Render unto Caesar...".
Now, contrast with Islamic teachings. Not every Muslim will advocate for Sharia, but there is a non-negligible part of them (leadership included) who think that not advocating for Sharia is a sin.
What’s the contrast? In both cases, there are good people who understand that their religion restricts them, not others, and there are bad people who think the government should enforce their religion.
What part of Christianity actively promotes it? There’s that one line, which meaning is debated, in a book full of stories about religious governments.
In any case, I only care about the practicalities. In terms of what they try to achieve, there’s no real difference between the Christian and Muslim dominionists.