I'm just guessing, but I would think that with a cheap system that the included sub/speaker may not have much of a crossover in it, but will allow the designed A/V receiver (Sony Dream System) to handle those duties digitally.
To check, if your current A/V receiver has an internal crossover frequency setting, you can play with that setting and see how it affects the audio you are listening to.
Realistically, you shouldn't expect much with a low end sub, especially one pulled from some other system.
As the above posters stated, this isn't something to put one penny into, and it just makes more sense to work on your own speakers as soon as money allows. Live with what you have.
As a guess, I would expect movie mode to add a boost of bass to the output, while music mode will not add any boost to the bass so it sounds smoother. If there is an internal crossover, it would likely not be affected by that button. The button is a replacement for a 'volume' control.
If there ARE internal crossovers, then they probably are set for the range that speaker can handle, which likely isn't much.