In general, people always care about floating point errors. However I disagree with Andrej, and I do not think that floats are preferred to arbitrary precision reals (for the most part) because of sociological reasons.
I believe the main argument against exact computation of reals is one of performance. So the short answer is, whenever performance is more important than precision, you'll want to use floating point numbers.
The application that springs to mind is the use of computational fluid dynamics to design the aerodynamics of cars or planes, where small errors in computation are easily made up with the astronomical gains of using dedicated floating point units found in many widespread processors.
In particular, the problem of representing a wide range of real numbers using a fixed number of bits is not as trivial as it may seem at first glance. In numerical simulation, values may vary widely (e.g. when there is turbulence), so fixed-point computations aren't appropriate.
Even when precision is not fixed by the hardware, using arbitrary precision numbers can be several orders of magnitude slower than using floating point numbers. In fact, even in the nice case were all the numbers are rational, simple operations like inverting a matrix can result in large, hard to control denominators (see here for an example). Many large linear optimization packages use floating points with appropriate rounding modes to find approximate solutions because of this exact issue (see for example, the majority of programs found here).