Most conventional amps have a power supply that will decrease in voltage if driven with a lot of loud bass. If you highpass that amp, you virualy increase the supply voltage. Which gives the remaining frequency range more headroom before clipping. There may be more advantages, like less heat produced and semi conductors working more linear, but this is the main reason for better sound from the amp.
The midwoofer will do less excursion, resulting in a better midrange reproduction, passive components see less current. All that leading to less distortion not only at elevated levels.
So not only does your sub give extended bass response, but it also reduces non linearities of the main speaker. Makinge them sound like some more expensive version of themselves.
The average main will sound audible better, high passed with a subwoofer. Using no high pass with a sub is simply making the investment less audible value.
Now, if you only add a sub and still feed the main speaker full range, if you get the settings right, have a very good amp and limit the SPL, this may come close. As soon as you crank it up or your main amp is more of a mediocre design, you will hear quite drastically what I described. Increased level will muddy the sound, just like before, just with more low impact. Often very ugly, because the crossover of woofer and sub doesn't fit. These are the people that tell you they don't like sub woofer, which is simple nonsense. If you are driving a bicycle and get run over by a car, you may not like automobiles, but still use an ampulance for a quick trip to the hospital. Any 3-way is in some way a sub woofer combination. Just in a single cabinet. Some use sub's for each stereo speaker, what about that?
Such an installation with a sub and without high pass is the same as bi-amping. You only will not get much of an improvement, except for the low end the sub can deliver.
Maybe read opinions about bi-amping to get an idea.
