What changes (in sound terms) should I have to do (in any audio
software after recording) to get better sound?
There are no technical problems with your recording that need fixing. I'm afraid I'm the wrong person to ask about using Audacity effects as the only ones I need for my voice recordings are noise reduction and (level) normalization. You can experiment with the available effects or see if another sound design user can help.
I somewhere heard an
interface [is better] for recording. Should I buy one for recording?
An external audio interface may be useful with low-level signal sources such as microphones and instrument pickups but is unlikely to offer any advantage with an analogue line-level source such as an electric piano/synth. As you're already using the USB to Host feature for digital recording on your Yamaha PSR-E453 you won't get better than that. People listening via a phone are likely to be listening on headphones or earbuds.
Why [does] Rousseau's recording sounds good ...
I ran your recordings through a spectrogram program on my phone. (Spectroid by Carl Reinke on Android, listening to recordings played through a 2.1 PC speaker system. There is a local 10 kHz signal at -72 dB.)
(I'm an electrical engineer but not a musician.) The voice you're using, which seems to have a musical box quality to my untutored ears, has quite a thin spectral content to it (left hand of the picture) giving a rather thin sound.
The acoustic piano voice used for the Rousseau recording (centre of picture) has a broader spectral content giving a thicker sound. The pink parts of the waterfall pictures represent sound components. There are more of them in the Rousseau recording, giving a richer sound.
For comparison the right hand spectrogram is an instrumental section from the original Queen recording of Bohemian Rhapsody (all instruments).
Artistic decisions are for you, not me. If you want a sound closer to the Rousseau recording you could re-record the piece using an acoustic piano voice rather than the current voice. Or you could choose one of the other 756 available voices. But we don't need all music to sound like a Bohemian Rhapsody cover. You should make your own artistic choices and not worry about sounding like someone else's work.
Having listened to the recordings on a phone, headphones and a 2.1 PC speaker system I can't hear a marked difference between them, beyond the expected lack of bass on the phone. I wonder if you are too close to the subject to get a full view of the difference between the live sound and different playback equipment? Do you have a local musical colleague who can listen to you play live then listen to different playback equipment and give you an independent view of the relative quality of varying playback standards?
(The spectrograms below (upper section) plot log amplitude vertically against log frequency. The yellow trace shows the instantaneous plot (at the end of the recording) while the red trace shows the peak level during the whole recording. The lower waterfall section shows amplitude nominally colour-coded by the scale to the left (in practice pink = sound). The frequency axis is shared with the upper plot. The vertical axis represents about 20 s of time, oldest at the bottom.)