To give the answer to the question whether amps improve sound quality we should understand what they really do.
How do amps work
In fact an amplifier is intended to improve the transmission of sound waves from one device to another taking into account their different impedance. Converting the signal of an audio source to the power that may be perceived by the speakers/headphones an amp makes it more detailed, clear, and loud. When we understand this process we get that we will better perceive the sound the way it is recorded, but it will not improve. The amp improves not the recording quality, rather the way of its reproduction. That’s why professionals use amps for more detailed sound to hear all the details unavailable to perceive without amplifying.
The sound quality and amps
Thus amps are great to make devices more compatible and get more sound clarity and volume, but not the quality of recording itself. The sound quality also does not depend on an amplifier only. You will not get a better sound of headphones or speakers than they are to reproduce. But an amp will squeeze a maximum of their possibilities to deliver the sound. So an amp will help to drive headphones to their full potential though will not make a miracle if they are not designed to reproduce a decent sound quality you expect.
In conclusion, the sound quality first depends on the quality of recording itself and the characteristics of your audio devices. And an amp will help to get the maximum of their possibilities and music details.