Spoiled by the power of your best tools, you tend to shy away from messy calculations or long, case-by-case arguments unless they are absolutely unavoidable. Mathematicians develop a powerful attachment to elegance and depth, which are in tension with, if not directly opposed to, mechanical calculation. Mathematicians will often spend days thinking of a clean argument that completely avoids numbers and strings of elementary deductions in favor of seeing why what they want to show follows easily from some very deep and general pattern that is already well-understood. Indeed, you tend to choose problems motivated by how likely it is that there will be some "clean" insight in them, as opposed to a detailed but ultimately unenlightening proof by exhaustively enumerating a bunch of possibilities.
What is it like to have understanding of very advanced mathematics, the rest here (ht Tyler Cowen at MR)