I am not a mathematician, but sometimes I get accused of being one; so given that no real mathematician have answered, I guess I can give it a shot.

Mathematician are in charge of building mathematical tools that are used by physicists, computer scientists, and many other subjects, including artist.

Why is math useful: mathematics are used in social science, physics, computer science and many other subject. Take a simple example from computer science: everyone is very excited about quantum computing, but what questions can be answered faster by a quantum computer than a classical computer? This is both a computer science question and also a math question. Many mathematicians are working on problem like these.

What is the difference between mathematican, computer scientists, physicist, and so on: although people from other subject also use advanced mathematical tools and work on similar questions as mathematicians (I guess why I was accused of being a mathematician), the difference is in their approach. Typically, for non-mathematicians (like me), proofs and math tools are means to an end. We often want to prove a very concrete problem (like are two reasonable ways to define the meaning of a program are equivalent), and usually we prefer the proof the takes the least amount of effort to get to the conclusion. Whereas mathematician often makes connection between different approaches, generalize, and just explore things that they feel is interesting. The mathematical approach often is slower but also gives deeper understandings: although it is common for many of their insights to be lost through time, it is also quite often for these exploration leading to important breakthrough in other fields.

What is the life of a mathematician like: like every other academic: teaching, research, writing grant to feed yourself, and sometimes traveling to discuss ideas and start new projects. I imagine OP is most interested in is mathematical research. I feel the most apt analogy is the creation of art: for an artist, they usually have a emotion trying to express, either something they see or feel. Then they do a couple sketch, see what detail/style works in expressing their ideas and what doesn’t, then paint the painting. For mathematicians, they often have a question in mind, then they try some examples to see what steps closer to their goal and what leads to dead ends. Through these excersices they gain a intuition of what conditions are important for the desired conclusions, then they pain the full painting by finishing the proof.

These proofs can be exceptionally time consuming: even for computer scientists, they can easily take couple researcher a year of work to do a proof. Most of the sketches will be thrown away, either because they are too convoluted or because they don’t lead to the correct conclusion. Usually, a proof by computer scientists like me can easily take 20-30 pages to explain properly, if not more; and the proof that were thrown away can double that quantity. I can only imagine proofs for mathematicians will be even more energy consuming.