Mathematics is independent of experience. “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?” So Albert Einstein said. He, of course, had degrees in both mathematics and physics.
Science is a discipline or set of disciplines involved with learning about the universe in a systematic way. The word science also refers to the knowledge gained in this search. Science relies on the scientific method.
The method makes science what it is. Ideally, the scientific method gathers evidence through observing and/or experimenting, and uses that evidence to test hypotheses. A hypothesis is a proposed explanation of how something works.
Experiments test hypotheses. The hypothesis is formulated first, and it must ask a question to which the experimenter genuinely does not know the answer. The hypothesis is often expressed as a mathematical model, but not because the scientist is a mathematician.
In theory, experiments must be repeatable, by other scientists using the same methods. The search for evidence must be unbiased, and experimenters must be willing to share their results, in order to advance knowledge.
Scientists experiment, and gather data. Mathematicians conjecture, and follow chains of logic. That is, they seek out patterns, by applying logic to axioms. Some axioms are statements that are assumed universally true, while others are statements postulated to hold true for a given set of values.
In either case, by definition axioms are not subject to experiment. They are the foundation of mathematics, which is thus a structure of abstraction, striving to discover the ramifications of what it assumes to be universal laws. Mathematics is infinitely useful, and particularly useful to science, but it is not science.
Science tries hard not to make assumptions, and to test every idea it comes up with about how things work. Mathematics is based on assumptions. It may be perfectly obvious that the assumptions of mathematics are valid, but it is not therefore perfectly true.
Mathematics is useful, and many of its leading lights say that it is beautiful, though that loveliness may not be obvious to those who struggle with adding fractions.
Science is also useful, and has produced beautiful ways of understanding the world. Both scientists and mathematicians share a demand for rigor, willingness to work, and a desire to get to the heart of the way things work.
Many people call math a science because it shares the rigor and some of the goals of science. Bertrand Russell, the English mathematician and philosopher, spoke of math this way, “Mathematics is the only science where one never knows what one is talking about nor whether what is said is true.”