Why I replaced the AT2020
After years with an Audio-Technica AT2020, I wanted a microphone that offered a clearer improvement rather than a slightly different flavor. I chose the Lewitt LCT 440 PURE and compared the two on noise, frequency balance, distortion, sound-pressure handling and voice.
Sound and noise
While the AT2020's sound is average, the Lewitt performs better in the low frequencies. There is also a noticeable difference in the high frequencies, but not in terms of volume - rather, it has a smoother and more pleasant sound.
Another difference between the two microphones is the noise level. In weaker signal recordings, the Lewitt produced better signal-to-noise ratio. When the signal level is set properly, the difference may not be as noticeable, but the Lewitt was noticeably clearer.

Distortion and sound-pressure handling
Another advantage of the Lewitt is that its THD value is half that of the AT2020. Even among the most expensive microphones, some have high harmonic distortion. The Lewitt is surprisingly good.
I tested how it reacts to high sound pressure as well. While the AT2020 distorts significantly above a certain threshold, even if the signal level does not exceed 0 dB, with the Lewitt I could scream and the signal remained clean. This is particularly important for singers who sing powerfully from up close. This result is probably due to the fact that the Lewitt has a large diaphragm, while the AT2020 has a small one.
The voice test
I tested the two microphones on a female voice. The difference was huge. The Lewitt added the extra something that I needed in the signal.
Conclusion
All in all, I am very happy with my new microphone and I can say that there has been a significant improvement in quality. This microphone is still in the relatively cheap segment of studio microphones, but I definitely recommend it. I have been dealing with vocal recordings for several decades, so I am not just speaking without a reason when I say that this microphone is a world champion.
Perhaps I should mention the Lewitt LCT 240 microphone as well. I listened to it a lot and came to the conclusion that it is not as big of an improvement over the AT2020. So if you really want to be happy, save up for the 440 instead.