How are these fields related?
I have been fortunate to have had exposure to these fields - to mathematics as a student, and to engineering through a career stint as a software developer. Though I have not been seriously involved with science, I’ve naturally had exposure through the other disciplines. Despite working in these fields for many years, only recently have I thought more about the relationship between them, and perhaps it might be useful to share my line of thought.
To discern the relationship between these fields, the first thing to consider is that each of these fields is related to the world in some way. By world I primarily mean the physical world (as opposed to, say, the conceptual world, which would be another interesting discussion). So a good starting point might be to better understand each field’s relationship to the world.
The goal of science primarily seems to be to explain the world. That is, to give reasons as to why certain phenomena occur the way they do. To give reasons for phenomena, one must be able to describe those phenomena, so description and classification must be part of science as well. Also we must note that the value of science is its ability to generalize - if each phenomenon were to have its own unique explanation it would not be much more interesting than having no explanation at all. So science develops principles that apply to general cases, and the corollary of this is another valuable feature of science: the capability to predict phenomena.
What, then, is engineering’s relationship to the world? It seems that while science aims to explain the world, the primary goal of engineering is to manipulate it. The first thing to note is that, in order to manipulate the world, it is easier to do so if one understands something about it. If I am to build some structure made of stone, it would be helpful to know properties of various stone types, e.g. type A is more durable than type B. So here we see a fundamental connection, that science enables engineering. But one quickly perceives this relationship to go both ways - if we improve our ability to manipulate the world, we increase our ability to observe phenomena and explain them.
Also worth mentioning is that to manipulate the world one may require more than just material - also important are tools, people and organizations, or processes, and science can be involved in all facets. But explanations provided by science are never entirely complete (and often up for debate), and engineering bridges the gap between explanation and action.
Finally we come to mathematics’ relationship to the world. One aspect of mathematics is providing formalisms to help describe the world and to communicate ideas. (One could also use the words “measure” or “quantify” instead of describe). If I want to describe a phenomenon like atmospheric temperature or orbits of planets, it is helpful if I have the concepts of “number” or “ellipse.” The relationship can also be mysteriously deeper than anticipated - if mathematics provides formalisms which describe the world, and establishes relationships between those formalisms, sometimes those relationships help explain phenomenon in the natural world as well. It also goes both ways, and study of the natural world can expand mathematics - a notable example being the development of calculus as a result of investigations into physics.
Of course almost every mathematician (and myself) would argue that the field is much broader than this, and has a deep and interesting relationship with the conceptual world, but for the purposes of our discussion we won’t delve into that.
So by considering these fields’ relationship to the world, we have begun to perceive certain qualities - those of description, explanation, and manipulation. What’s interesting is that these features seem to be intertwined in solving any problem, or perhaps in any human endeavor. To solve a problem, it is best to define the problem, to understand the problem, and then to apply that knowledge in order to skillfully solve the problem. And while this categorization seems formal and distinct, perhaps that need not be the case - in art (or any activity), one must have some idea to express, an understanding of how to express and a means to express it, but one might do these things simultaneously, or without differentiating between them.