I am currently learning piano by myself. So far, after 1 year and 3-4 months, I I can play some hard pieces. But my piano is unweighted one, and things become different when I have to play on a classical piano. Slow pieces are OK (Mozart's piano sonata no.16 - 2nd movement, Beethoven's "Pathetique" - 2nd movement, etc. or even Chopin's Etude No.3 - Op.10). All I need to do is focus on dynamics and pedal. But when it comes to fast pieces like Turkish March, Chopin Etude No.4 - Op.10, etc. or even Fur Elise, I simply can't play like I used to. My fingers don't push hard enough. If I can overcome this problem, I'll have opportunity to be a good pianist (I sight-read very well, can improvise in any key just in a blink, my knowledge on music is good (not only chords and keys but also counterpoint; I'm a true classical composer, like Beethoven's time)). So, my question is: Is there any way I can overcome the difference between weighted and unweighted piano, without having to practice on a weighted one?
Put simply, no. If you've got this far and this good on a non-piano feel keyboard, you must owe it to yourself to do the job properly, and re-learn on a 'proper' weighted keyboard, be it a real piano, or a very good electronic keyboard, with weighted or at very least, semi-weighted keys. It's much more than merely increasing finger strength, though. The whole concept is different, and it will take a while to accommodate the different feel and techniques involved, and the pedal action will not be the same either.
Since you mentioned "true classical composer" then it might be worth mentioning clavichords, harpsichords and organs. In the baroque era those were basically the options for keyboard instruments. People couldn't well have giant organs in their homes so they had to settle for clavichords/harpsichords. Harpsichords were still quite large and expensive so you would have to be wealthy enough to afford one and also have enough space to play and store it. Clavichords were basically the equivalent of the "unweighted cheap keyboards" that students have to use these days.
Keyboard players would then practice and compose using the cheap light limited instruments and perform on the larger more complex organs. Organs don't have weighted keys but they are still plenty more complicated that clavichords.
However no matter how skilled they got on their practice instruments they would still need to practice on the organs in order to raise their proficiency at it.
Unweighted are fine and fit for their purposes; cheap keyboard for beginners or those who can't afford to practice with. It is better to have one than no keyboard at all. You cannot unfortunately overcome the differences without practice time anymore than you can practice truck driving in a car. You can drive in a car and accomplish things but a truck is a different thing that you just can only improve on with direct experience.
I second the NO answer. I play both weighted piano keyboards and unweighted organ keyboards. To some extent you can play them interchangably. But when you get down to playing them well, they are different and you must have the right keyboard for the type of music being played.
Piano music requires that you control the velocity of the key to control the dynamics. Doing so relies on the inertia of the piano hammer mechanism. You can't learn that control without a weighted keyboard.
Organ music requires holding keys for the entire time you want the note to sound which can require a lot of finger substitution and "crawling" over the keyboard. Organ technique relies on the fact that you do NOT have to control the velocity of the key. You can use finger movements that are disadvantaged for apply a forceful movement of the key. It is more difficult to play organ music on a weighted keyboard than an unweighted keyboard.
So you have to have the right type of keyboard for the type of music you are playing. The good news is that it is nowhere near as expensive or space consuming as it used to be.
First of all, any good pianist should have a weighted keyboard; whether it is a good electric piano, a synth like Korg Kronos or a home grand piano is up to you!
The difference in playing on a weighted keyboard and an unweighted one is significant.
I am a keyboard player myself and I can say I started learning on an Yamaha DGX-520 and it wasn't a weighted one then when I started playing a Korg Kronos in a band I had a lot of trouble with fast licks and playing organ. So I played it a lot and my fingers got stronger and I felt how faster I've become.
After you've played a weighted keyboard once you sit down to a simple synth that does not have that you feel more powerful than ever, your fingers really start flying and everything becomes much either, you can't properly feel the unweighted keyboard.
The other good thing to remember is the proper work of the body, arms, hands and fingers while playing. I hope you had a teacher who explained that when you play you should use the weight of the whole body to take the note or chord you need. There are many lessons to learn that.