I'm trying to play this piece, there's no sign to use the sustain pedal in the sheet music, but when I play it and compare it to the original track, I feel that the sustain pedal should be used. I am just confused whether this sheet has some problem or I'm playing wrong legato sections.
Not all composers write pedal indications in their scores. If it's really important they may do so (or write "sec" or some other word to indicate no pedalling). Rachmaninoff was noted for excellent pedal work in his performances but he didn't write much in his scores.
To some extent, it's up to the performer. The same piece may need different approaches to pedalling in different venues. A venue with long echo time (cathedral) may need little pedalling where as a Texas country-western bar with wood and sawdust floor may need more. (A similar problem arises with tempi of pieces.)
The "If it sound good, it's correct" can be a reasonable guideline.
Pedalling is fine when all (or most) of the notes played while the pedal is down blend together, and it's cancelled before the next lot. So here, in this piece, with each separate bar fitting to a particuar harmony, it'll work well. The melody is quite static, so ther's not going to be notes bleeding ito other notes, and the triads underneath are simple sounding chords. It will work well, for this piece.
Using pedal also means more notes are heard - the harmonics of the played notes ring out too, giving a richer sound. That may well be the difference you hear.
As long as you're playing legato, and sustaining the left hand's dotted minims correctly, the first 50 bars wouldn't benefit much from pedalling. But in the next section, where the left hand moves in quavers, it needs some indication of how it's to be played. Even 'senza ped. or 'staccato' or 'legato' would have been helpful.
I don't know the original, but if it sounds as if a pedal's being used I would use one if I were you.