I just finished G. H. Hardy's celebrated "A Mathematician's Apology". He describes "useful" math such as calculus as "on the whole, rather dull", and contrasts it with the "real" math of Euler, Fermat, and Gauss, which he finds "almost wholly 'useless'"; he adds that the great achievements of applied math--relativity and quantum mechanics--are "almost as useless as number theory".
He was prudent to add "at present [1940], at any rate": the phone on which I am typing wouldn't work at all without number theory, relativity, and quantum mechanics!
I have a vague sense that mathematicians take some pride in the uselessness of their discipline, even when it's the exact opposite of useless. Maybe because the more useless the mathematics appears to be, the more it must be motivated by the pure love of the subject?
With regard to the "pride in uselessness" thing, I caution you against generalizing here. Different people get into math for different reasons, but I would say that all the pure mathematicians I know of are motivated principally by interest in the problems they care about, and don't think too much about questions of purity.
There's a nice quote from Courant in his "Introduction to Calculus & Analysis" where he warns against "smug purism", exhorting students to draw inspiration and insight from other fields because it will make them better mathematicians. I think this is the attitude that I encounter most frequently among mature, pure mathematicians.
Chebyshev in particular was known not only for working on problems that had engineering applications, but for using methods and techniques from engineering to inform his approach to pure math problems. As the founder of the St. Petersburg school of mathematics, this approach had broad impacts on (later) Soviet mathematics and global mathematics as students brought up in this tradition went on to train later generations of mathematicians around the world.
He was prudent to add "at present [1940], at any rate": the phone on which I am typing wouldn't work at all without number theory, relativity, and quantum mechanics!