Mathematicians Vs. Computer Scientists

I always had this confusion in my mind about what Computer Scientists do. We've mathematicians who are usually associated with design of algorithms and than we have electronics wizards who are designing computers of today and tommorrow. This thinking always confused about relevance of Computer Science as a subject. However while reading an article about Parsing, I came across this note by Authors (Dick Grune and Ceriel J.H. Jacobs).

"There is a considerable difference between a mathematician's view of world and a computer scientist's. To a mathematician all structures are static: they have always been and always be; the only time dependence is that we just haven't discovered all of them yet. The computer scientist is concerned with (and fascinated by) the continuous creation, combination, separation and destruction of structures: time is of essence. In the hands of a mathematician, the Peano axioms create the integers without reference to time, but if a computer scientist uses them to implement integer addition, he finds that they describe a slow process, which is why he'll be looking for a efficient approach. In this respect the computer scientist has more in common with physicist and chemist; like these, he cannot do without a solid basis in several branches of applied mathematics, but, like these, he is willing (and often virtually obliged) to take on faith certain theorems handed over to him by mathematicians. Without rigor of mathematics all science would collapse, but not all inhabitants of a building need to know all the spars and griders that keep it upright. Factoring off certain knowledge to specialists reduces intellectual complexity of a task, which is one of the things computer science is about."