When you filter a signal (with a standard low- or band-pass filter)), you indeed produce two "subbands": the one you keep, the one you leave out. The filtered signal might have less noise or unheard components. Its amplitude might be somehow a little smaller. But it keeps the same number of samples. So its is not really compressed in a bitrate sense.
Lossy compression is about saving as much bits as it can: either on amplitudes and on the number of samples. First, subband filtering (multiple filters in parallel) allows you to filter the data into say 32 bands, and each can be subsampled by 32, almost without loss of information. So you have split the signal into 32 simpler ones, at almost the same bitrate. Now you can reap the benefit: each simple subband signal is closer in shape to a sine, that is easy to predict. Moreover, in each subband, at each time, you are able to exploit fine auditory properties:
- a weak sound that closely follows a strong sound is not heard (you can save bits by not coding it),
- a weak frequency close to a strong frequency is not heard (you can save bits by not coding it),
- depending on the frequency range, your ear cannot distinguish easily between close frequencies, or amplitudes can vary by 15 % without getting noticed (and you can save bits in coding the amplitude less precisely).
All these tricks can be used because the signal is in time, and your subband filtering splits it into fine frequency bands (a time-frequency representation).
By contrast, with a single filter, you tend to treat all time instant equally, while it is beneficial to treat them different depending on the context (strong amplitude or frequency close-by).