For a very long time, I wanted to be an illustrator before I got interested in engineering and then eventually studying math (finished a PhD) before my current job, which is software development.
Depending on what kind of math you do, visualization can be very important, but I found a lot of mine was essentially 2-dimensional, even when I was thinking about n-dimensional things. However, it's not clear what the link is between drawing and math visualization and what skills would really transfer over. I never developed my drawing skills quite to a professional level, but I got far enough that I think I really understood the essence of what drawing is about, such that a pro mostly just does the same things I do when I draw, only better.
But some of the most effective techniques involve mentally ignoring 3-D space. For example, there is the "Drawing On The Right Side Of The Brain" approach that emphasizes "negative space" (in the sense that such terminology is used by artists). That approach urges the artist to forget the 3-D mental model of the subject. Drawing "negative space" applies to a situation where you are drawing something you actually see rather than drawing something you imagine.
This is a different kind of drawing than the sort needed by, say, a comic book artist, or in other words, the ability to draw from your imagination. If you just copy what you see, you only develop fairly superficial drawing skills. Even to draw more artistic life drawings, you have to think somewhat 3-dimensionally. Drawing is about as sophisticated a skill as being a concert pianist or getting a PhD in math. The whole book Drawing on the Right Side of the Brain is like dipping your toes into the ocean of drawing.
It turns out that the sense of touch may be almost as important for drawing as vision is. Conceptually, I've always thought of learning to draw from the imagination as sort of gaining possession of a kind of imaginary clay that you can work with on paper. I think this sprang from my observation that sculpting was quite a bit easier than drawing was--that gives you a sense of how touch might come into play when creating 3-d images. It's not clear that this any of this is really that helpful for doing math, but you can always check out a copy of The Natural Way to Draw and working through the first chapter (or more, if you are so inclined) if you are curious to experience a taste of what I am talking about.
It's possible that there could be a benefit to your geometric reasoning abilities, and all you have to do is remember the existence of projective geometry and even regular plane geometry to see some connection between the two subjects, but I am always a bit skeptical of whether skills really transfer from one domain to another. It's hard to say, but I think the skills are fairly independent of each other.
I think I may have been too visual to fit in in today's mathematical culture, even after choosing a fairly visual subject (topology), so the imagination is probably not held in quite the esteem it deserves to be in math, despite the fact that many of the very best mathematicians have stressed its importance in their thought processes.
