What is the purpose of dividing a signal into subbands when compressing an audio file?
We don't hear all frequencies equally well. Therefore, if we split the frequencies into subbands, we can give more bits to bands which we hear well, and less bits to bands we don't hear that well (which is determined by things like doing listening tests). Its usually easier easier to control the error of the compressed signal on a subband level than on the whole signal level. 

This will typically give us less distortion for a given data rate than trying to encode all frequencies with equal number of bits.

An analogous experiment you can do is with (natural) images: Take the DFT or DCT of an image, and throw out the higher frequencies (i.e. high in the x,y directions). You still get a pretty decent approximation if you keep only a few low frequencies. So, instead of throwing them out, imagine you quantized them -- you'd want to give less bits to the higher frequencies than the lower frequencies, since you're not getting much perceptual gain by throwing lots of bits in the higher frequencies, but you are getting more in the lower frequencies. (This underlies JPEG; See something like Gonzalez and Woods' Digital Image Processing for details on how exactly JPEG works).