When I first started working out, I wore anything, and by "anything," I mean shorts and a cotton T-shirt. Over the years, I've started dressing for the specific activity (running clothes for running, cycling clothes for cycling / spinning, yoga clothes for yoga, etc.) and the reason isn't so I can single-handedly support the makers of athletic apparel.
Take running and cycling (or spinning), for example. You'll notice there is a significant difference between running and cycling tops (for women): running -- and other tops for random cardiovascular activities as well as your everyday tank tops -- are designed for the wearer who remains upright, while cycling (or triathlon) tops are cut very high at the neck, providing coverage for the user who is bent over handlebars. On the left, a running top, on the right a cycling (technically a tri) jersey:
It doesn't take much of an imagination to figure out what happens when a woman wearing anything but a top cut high at the neck leans over. In spin class where there are mirrors around the room, it doesn't, in fact, take any imagination at all.
Which brings us to shorts. When you're riding on a saddle that's getting up in there (and that's been up in a lot of theres as is the case in spin class), you want a little something between you and the saddle. Thus, bike (or tri) shorts offer a lot more coverage on the thigh compared to running shorts -- good for preventing chafing as well as separating church and state -- and they're also fairly high cut on the back so they don't gape and leave you with a breezy derriere as you lean over. Running shorts on the left, bike shorts on the right:
Lastly, shoes. Granted, a 45-minute spin class isn't exactly a stage of the Tour de France, so footwear isn't of paramount importance, but a stiff-soled cycling shoe will still give you a more efficient pedal stroke than a (relatively) floppy-soled sneaker. (Again, running on the left, cycling on the right.)
Also, cycling shoes clip in such that the pedal is under the ball of your foot, which generates more power than if the pedal is under your arch. Many people erroneously jam their sneakers all the way into toe cages so that the pedal lands under the arch of the foot, not realizing that it's more efficient to pedal with it under the ball of the foot. (Incidentally, the cycling shoe above is a mountain-biking shoe because I left my spinning shoes in the office. Because MTB shoes are quite heavy, most people prefer to spin in road or triathlon shoes -- not that it matters all that much since you're not exactly going anywhere on your spinning bike.)
In New York City, stores like Jack Rabbit and Paragon sell both running and cycling gear. Paragon also sells their goods online and Sierra Trading Post also sells a lot of running, cycling and triathlon gear at a discount, although they are absolutely the king of junk mail, so if you order from them, be prepared to get a zillion catalogs.