Richard Feynman was obviously an extraordinary physicist and a charismatic lecturer. In the following video he discusses the differences and the interplay between mathematics and physics:
His opening statement is: “Mathematicians only are dealing with the structure of the reasoning.” Actually, as many other mathematicians, I believe that dealing with the structure of the reasoning is a very secondary (and boring) aspect of my work. From a personal perspective, I think that, ideally, research in mathematics is more about creativity: establishing connections between ideas and concepts based on a criterium of elegance and beauty.
The psychology of the invention in mathematics has been adressed by one of the greatest French mathematician, Henri Poincaré, in the following famous lecture before the Société de Psychologie in Paris: Mathematical creation.
This lecture inspired the beautiful essay by Jacques Hadamard: The psychology of invention in the mathematical field, that I recommend to anyone interested by the research in mathematics. A very interesting discussion is also given by Paul Halmos in the following paper: Mathematics as a creative art.