There are at least three things wrong with the idea that they don't.
The view that science and religion don’t mix is guilty of at least three logical errors.
First, it commits the either/or fallacy by asserting that a view is either scientific or religious. Design models have some evidential support. For example, we see the blending of science and religion in the existence of a Creator based on Big Bang cosmology as the beginning of the universe.
Second, it commits the straw-man fallacy by assuming that creationists make no use of scientific methods. This is not the case. Creationists are happy to present a ton of scientific evidence for their view, if they're allowed. This evidence needs to be addressed instead of disqualified.
Third, it assumes that the reigning scientific views do not have religious significance. This is false. All cosmological views have religious significance. If evolutionary naturalism is true, the only place for God is in the imaginations of believers.
A clear hard line between religion and science just isn’t possible. Instead, they should work together, drawing on the strength of each to give us a total picture of reality.