iternabe I just played a little of the piece on my acoustic piano.
If you press a key, that key's damper is lifted off the strings, allowing it to resonate.
If you press the regular sustain pedal after you press a key and before you release it, all the dampers across the piano lift, and that one note is sustained, with a little bit of sympathetic resonance.
If you press the pedal to hold the first note, any other note that is struck sustains, and gets the full resonance of all the strings in the piano. It sounds bigger, fuller, and more resonant. This is why you need to release the pedal and press it again immediately when you change chords, to prevent dissonant chords. If you don't pedal at the marks, you could get a mushy sound or uneven sound.
The pedaling mark is telling you where the composer arranger/editor wants which kind of resonance from the instrument.