The New York Times has an interesting article arguing that imprecise approximational intuitions--the ability to quickly and roughly reckon things--are crucial to success at abstract mathematics.
To someone whose mathematical work was in real-number based areas--analysis and probability theory--this is very plausible on introspective grounds. But I wonder how true it is in more algebraic fields. I've never been very good at higher algebra--things like the Sylow theorems were very difficult for me (I still passed the algebra comp, but it came noticeably less naturally to me than the analysis comp), perhaps in part because my approximational intuitions were close to useless.
Anecdotal data suggests to me that there are two distinct kinds of mathematical skills. There are the skills involved in analysis, skills tied to problems that are real-number based (complex numbers are real-number based, of course, since C is just the cross product of R with itself), often visualizable, and where approximation and limiting procedures may be relevant. And then there are the skills involved in more algebraic fields, where (as far as I can) approximation gets you nowhere, and while visualization is helpful, the visualization is much more symbolic (visualizing a path of a brownian particle is pretty straightforward, one visualizes quotient groups either explicitly in symbols like "A/H" or perhaps in some strange and highly abstract diagrams). I don't know where to put the combinatorial--it may somewhat straddle the divide (a lot of visualization is involved), but I think is very algebraic in nature.
It is quite possible for a person to be really good at one of these, without being very good at the other. There are fields of mathematics that call upon both sets of skills. And there is an asymmetry: I think the analysis-type skills may be of very little use to mathematicians working in very algebraic areas, but just about every mathematician working in an analysis-type area needs to be able to do algebraic manipulation (though I have a strong preference for proofs in analysis-type fields where the algebraic manipulation is just a way of making precise what is intuitively obvious).