I recently read Timothy Gowers' essay "The Two Cultures of Mathematics." Gowers attempts to dichotomize pure math into two groups, "theory-builders" and "problem solvers." I think he is unsuccessful in describing a genuine mathematically-motivated difference between these proposed classes. I do agree that there are sub-cultures within mathematics, and that sometimes these sub-cultures criticize each other. And I also agree that this criticism is mostly fruitless and that the community would benefit as a whole if different cultures attempted to learn from the ways of other clans rather than to judge them. Gowers believes that these two cultures derive from differences in the way folks approach problems and he argues that in certain key ways, the two cultures are not all that different and should respect each other more.
"Theory-builders", as he puts it, are people who are mostly interested in providing proofs of results that yield purportedly "deep" connections between "core" branches of mathematics. Core branches are meant to be things that ultimately apply to "all of math," like the way general topology subsumes much of analysis, and the way group theory intertwines with geometry. Meanwhile, "problem-solvers" are people who concern themselves with a specific statement of a specific problem (like a specific bound in a graph coloring problem). While such problems might ultimately be "vindicated" by having deep connections to other parts of mathematics, the right-here-and-now reason why a problem-solver attempts to solve it only has the narrow scope of the satisfaction of having solved this particular problem as-is. Gowers uses the term priorities to describe this difference between his suggested two cultures: theory-builders prioritize abstract connectivity while problem-solvers prioritize narrow-scope utility to resolve a specific problem statement.
Gowers argues that "problem-solving" in this narrow-scope sense shouldn't be regarded as somehow inferior to theory-building. It's just that problem-solving develops theory in a much different way, normally as a set of heuristic methods for solving problems. Then the heuristic methods are fleshed out and potentially generalized and packaged together to yield something that might be satisfactory to a theory-builder, and might offer connections across a wide range of problems in different sub-cultures of math. I think it is a detriment to the essay that Gowers focuses exclusively on illustrating this through his own field of experience, combinatorics, because the litany of combinatorial examples make it feel like he's building a case for his argument, but really they just distract from the semantic game he is playing with what is meant by "problem solving."
My own belief is that the two cultures Gowers speaks of have come about because of the ever tightening economic constraints on the field of pure mathematics and career politics of making a career as a mathematician. It used to be the case that the aesthetic grandeur of a math result and its connections to theory-at-large actually were the things that a career mathematician was commissioned to discover. But in the modern research and university setting, this will hardly do. So on one hand, you have many people in the "problem-solver" group who are able to keep a job if they agree to work on problems that eventually have a basic practical interest, like graph colorings or sphere packings, or esoteric cryptographic algebra and number theory. By winning a grant that's partially predicated on discovering some practical cryptographic result, these "problem-solver" types are often able to dawdle around with combinatorial research on e.g. graph coloring bounds, and it is one of the few ways a person can make a career out of the mental mathematical effort that they enjoy.
In almost no other branch of scientific higher education can you get away with this.
On the other hand, though, the problem is worse over in "theory-building" land. Why should anyone invest money into a grant that pays for mathematical theory building? Ultimately, the answer must be that it will have important practical consequences in the daily lives of human beings . Alternatively, a wealthy patron may want to commission a work of mathematical art, but that is such a rare way for math to be funded that it's not even worth considering here .
I think this all affects theory-building in the following way. Instead of patiently building a theory from the ground up, and solving combinatorial-esque problems that at first appear to be narrow-scope and only later open up to reveal methodological connections (the way folks used to do math, a la Gauss, Euler, Cauchy, etc.), modern theory-builders have to go right for the jugular. Most people will not even publish results until they solve the problem in as abstractly generalized a manner as they can, lest someone else picks up their publication and "bests" them by seeing some sort of more abstract view of their work. This is because the work is absolutely scarce, and the only things that seem to garner any kind of tenure-winning notoriety or fame are solutions that cut a hot swath right through all of the combinarotial-esque details of a problem and go straight to the super generalized view. Because this is one of the few ways to be rewarded as a pure mathematician, it's not surprising that people tend to spurn or pass judgement on other ways to eek out a math career, such as methodically solving narrow-scope problems that have connections to practical interests, and hoping to string together several such narrow-scope ideas into larger ideas .
Here is a more succinct way to put it. Career competition for the theory-builder (most mathematicians) is all about going for the jugular and never publishing anything that isn't outrageously abstract from the outset. Nothing would be worse than to be "scooped" on the very story that you broke yourself because you didn't see it in a general enough setting. Proofs are easy; there are way too many people trying to do proofs for a living. If you want to do proofs for a living because you think it's fancy and shows off your high-falutin intellectual vision / "mathematical creativity," then you have to push for ever more abstract results to be obtained ever earlier in the research process. Contrary to this, you can be a career mathematician if you are happy yoking yourself to narrow-scope, uncreative practical questions and then just ripping off the grant-awarding agency by shirking the practical application you're paid to solve and instead focusing on some semi-related mathematical curiosity you happen to like, and then hoping to package a few of those together into a bigger idea every few years.
Theory-builders seem to pass judgement on this second type of mathematician (of which combinatorialists are usually an example). In my view, this is not so much an issue of dramatically different approaches to mathematics as it is an issue of approaches to carving out an academic career within mathematics. One side is engaged in fierce struggles, trying to justify their existence with ever more rapidly-abstracted ideas. Meanwhile, the other seems largely protected by low-risk-low-yield "practical" grants that allow them to dawdle with their happy little puzzles. It seems like a problem of tribal association, and a problem of professing and cheering.
 Of course, there is always the classical philosophical debate about the usefulness of new theories. It is attributed to Faraday and Franklin that, when asked what good a new theory is, they replied "what good is an infant? You cannot know until it grows up." This might sound like a heroic defense of the pursuit of abstract theory, but it will simply not do as an economic incentive for taxpayers to be forced to hand dollars over to mathematics faculty.
 Although, I personally would love to see more things like the Clay Millennium Prizes and something like the X-Prize brought over into the math world -- and especially to target such prizes at non-faculty mathematicians who work on math out of hobbyist love rather than for a career.
 As an aside, I once heard a respected mathematician giving a lecture on how to organize work when trying to prove a difficult theorem. His advice was that you should go hash out all of the messy calculational details on some scratch paper somewhere, see "why" the proof succeeds, and then write up your publication-ready version of the proof in such a way that it hides all of the messy details from the reader but illuminates the "why" behind the proof's success. I cannot think of worse advice, for if you are a research mathematician, then the people who read your work only want to know how it is that you did all of those messy calculational details. They don't care at all about how smart your proof makes you look.