Science is about verifiable proofs (verifiable within reasonable time limits) which somewhat means the same thing as formal proofs in the grand scheme of things. Correct me if I am wrong
Edit: forgot to put in time limits
Edit: forgot to put in time limits
The various disciplines that generally rely on "formal proofs" are referred to as "formal sciences" (logic, mathematics, statistics, etc.), but are technically not actual "sciences" since they are fundamentally abstract (as opposed to how we defined science above) - which is why they are generally concerned with formal proofs and not empirical evidence. Of course the formal sciences frequently provide the natural sciences and social sciences with ways to describe the physical/natural world and the social world, respectively.
Science is about building and refining models (theories) in order to make them match observable reality as close as possible, in order to use those models to predict what happens in the future - both by itself, and in consequence of us poking stuff. That's what it means to know "how something works".
Engineering takes these models and adds a "what's the best way to poke things to achieve a desired outcome?" aspect.
Formal proofs are for mathematicians. Mathematics is a purely abstract invention and operates in its own universe, where absolute formal proofs are possible.
In your defense, science appears to follow formality, though we run into often run into holes in our theory when a confounding number of variables are in play.
Me, personally, I don't know what that difference really is. Science is science because it works. If it didn't work, it wouldn't be science.
Mathematics can and will find ways to apply itself to the real world, no matter how approximately, as long as that difference matters to somebody.
For example, a system like Coq (https://coq.inria.fr/) is concerned with formal proofs, but is not really a foundation for most of science.