I think you're right to have these doubts, that is to say, I have had similar misgivings about the "usefulness" of my work in pure mathematics, and not only that, but the "usefulness" of pure mathematics itself, and having pondered them for some years they have slowly ceased to be misgivings and grown into convictions.
This is not to say that there aren't differences in "usefulness" between various areas of pure mathematics, and it is also not easy to narrow down what "usefulness" should mean. However, I have never seen the fact that I can't define a word as a reason not to use it. Thinking in terms of definitions and axioms comes at the heels of intuitive thinking, even in mathematics, so all the more in philosophizing about mathematics.
For me, the doubts about the "usefulness" of pure mathematics were ipso facto proof that the word "usefulness" has a meaning, even if I couldn't give a definition. Now some may take this as a rather grandiose form of self-confidence, but they would miss the point. If I can only express my doubts by using the word "usefulness", then the word must have a meaning, because it is the only possible way to describe the specific form of doubt that I was experiencing.
And I am actually saying this by way of advice. If you force yourself, as you do, to not use a word because you can't define it, even though the word accurately captures your sense of doubt or dissatisfaction, then you are in effect preventing yourself from accurately describing and understanding your own problem. (Again: not accurately in the sense of every word being susceptible of a mathematically precise definition, but in the sense of being a satisfactory expression of your own emotional/psychological/spiritual state.) I know it can be intimidating to be among mathematicians and use a word in a vague, philosophical way, because they could ask you to be more precise, and then you probably feel you couldn't, and it would be as if you lost the argument or something. But it's better to be thought a little less of by some, if it means that you retain the freedom to express your thoughts in the way that they occur to you. This is a lesson that I had to learn the hard way, but I am glad that I learnt it.
However, it might be (as I glean from your post) that your conception of the term "usefulness" is a more utilitarian one. But I don't think utilitarian considerations can ultimately help you decide whether your pursuits are meaningful in a more profound and personal sense: the latter has more to do with your sense of usefulness, with "that which makes you tick", and there seems little a priori reason that this personal sense would be somehow perfectly aligned with the "greatest good for the greatest number of people". (Regardless of your stance on utilitarianism -- personally I think it is simply bad philosophy -- this would be a heck of an assumption to make.) This is especially the case because this "greatest good" is most often not something that is "felt", but rather an abstract reality, so not personally meaningful to the person who is pondering the question, but rather a dry and impersonal summary of things that contribute to people's general sense of well-being, and only up to a certain point.) And conversely, no amount of proof that activity X is conducive to the general good or happiness or well-being of mankind, could force you to feel enthusiastic or passionate about activity X, or to experience happiness while engaging in it.
In regards to my own doubts, I could be a bit more precise on what I could have meant by "usefulness", or rather by my feeling that pure mathematics felt insufficiently "useful" to me, but it is more of an illustration than a definition. What bothered me about mathematics was mostly the financial side of it. My salary was being paid by the government, which meant that I felt some sort of responsibility to be able to explain to the average person in the street why what I was doing was useful (not in quotes this time, but actually useful according to the definition of the average person in the street!). I had never been able to do this without somehow having the feeling that I was stretching the argument, that I was somehow selling a lie. And I had tried a lot.
But then the question remains: do your doubts about mathematics only reveal something about you, i.e. that you are temperamentally unsuited to do mathematics or something, or do they say something about mathematics as well?
I think the right answer is that I can't tell, because I do not know you. But let me once more tell you about myself. I think I am temperamentally inclined to like mathematics, even to be passionate about it. I have had generally good results as a one-on-one tutor, I enjoy conveying the beauty and the fun of mathematics, and I have often had people say to me that they were jealous of me for being so passionate about something. I think this pretty much seals the case for my "temperamental aptitude" for mathematics.
Yet at the same time, here I was, having these doubts about the usefulness of mathematics.
So I can't see this as a purely subjective issue, and I don't see it as such. Let me be a little bold, and put forward the following thesis. I think that some, maybe most, pure mathematicians have created an idol out of pure mathematics. Which is to say: they have somehow set it up als the ultimate goal of life, the greatest pursuit a human being can engage in, the search for the ultimate form of truth, etc. Of course, they wouldn't phrase it exactly this way, but it was definitely the kind of rhetoric that first got me into studying mathematics when I was in high school. Mathematics is the "queen of the sciences", it is the language that the book of Nature is written in, etc. The so-called Proofs from the Book are even ascribed to God himself, if you please.
Personally, I have often experienced a similarity between a "cool" bit of mathematics, and a "cool" new gadget such as the latest iPhone. (Interestingly, you also use that same word.) As with all "cool" things, they quickly lose their glamour after some engagement with them. When you first encounter a new bit of mathematics, it can have a veneer of the miraculous about it, but this is always lost upon closer inspection (in my experience anyway). There is a saying by Schopenhauer that touches on precisely this (I can recall it only very roughly): There are three stages in encountering a new truth, the first one is where it is not understood, the second one is where it is beginning to be understood and seems the most exciting thing in the world, and the third and final one is where it is fully understood, and is considered to be trivial and uninteresting. End quote.
So no, I have grown totally disillusioned with the claim, often made by pure mathematicians, that the pursuit of pure mathematics can contribute anything towards the good life. Now you might again like to enclose that last term within scare quotes, the "good life"; however, I think the term catches rather precisely what I myself am after when I am wondering about the "usefulness" of this, that or the other.
All this is not to say that pure mathematics can't still be fun. And insofar as "fun" contributes to the good life, mathematics contributes to it as well. I mostly experience this fun side of mathematics when teaching mathematics to high school kids and students, which I still sometimes do in my spare time. And if I manage to convey some of my enthusiasm to them, I often find myself warning them not to become enamoured with the subject too much. The key to being good at mathematics lies far more in precision, intellectual rigour and discipline (imho), than in imagination and speculative fervour. An overemphasis on enthusiasm and passion could very well obscure this fact, and steer a person wrong. As it did me.