Mathematics and science

What is the relationship between mathematics and science?

Not necessary, not close - as any perusal of the major breakthroughs in the history of science will reveal. 

But people who are good at mathematics, and who don't know much about the full range of science, like to assert that all "real" science is mathematical, or that maths is the key to science, or that maths expertise means you are a kind of arbiter of science...

Maths is one thing; science is many other things - some of which overlap with maths. 

In a nutshell, science makes models of bits of reality. Some models concern reality is structured  - these are qualitative models, usually made of categories - and are not mathematical - or only seldom. Some models concern how the world functions, and some of these are mathematical. 

Maths is also a kind of model, but not a model of the world but a model of mathematics, Some bits of this mathematical model can be used to represent bits of science: i.e. some bits of maths can become scientific models (which represent bits of reality).

Unless you are a special kind of Pythagorean or Platonist; here is no a priori reason why mathematical operations should necessarily be the same as all the functional operations of real life - and they aren't.

We do not live in a reality underpinned by the abstraction that is mathematics - but a reality underpinned by God, who is personal.  

So; both ultimately and in practice, the relationship between mathematics and real science is variable and not close; although maths (numbers and calculations, statistics and mathematical models) has and is often been the basis of the varieties of pseudo-science - an activity designed to manipulate people, rather than to understand the work.