The short answer is:this will probably never be determined by the inherent limitations of either medium. There are dozens of factors that influence the sound quality you get from a recording. These factors include your own hearing, room acoustics, the quality of your speakers, the quality of your equipment, the quality of the mastering job, the quality of the recording and the quality of the performance.
Frequency Response
CDs have an absolute upper frequency limit of 22.05 kHz. For all practical purposes the frequency limit is 20 kHz. The frequency response up to that frequency can be nearly flat though.
Vinyl on the other hand has no absolute frequency limit and frequencies up to 30 kHz or even higher, can be reproduced. In the past, there was even a quadrophonic system that used frequencies up to 45 kHz to modulate difference signals between front and back channels. This required special cartridges to play back and direct metal mastering, but it could be done.
What about your own ears? Current scientific consensus is that nobody can hear frequencies higher than 20 kHz and adults certainly can’t. People cannot distinguish signals that contain harmonics above 20 kHz from signals that don’t. Therefore CDs have a frequency response that is adequate for human hearing.
Note that FM stereo is limited to 15 kHz and people consider it superior to digital radio standards. So apparently, 15 kHz would be good enough for all practical purposes.
Dynamic Range
CDs have a resolution of 16 bits. In terms of signal-to-noise ratio, this is about 96 dB. You have a hard time getting more than 70 dB out of vinyl. So even if you consider the hard upper limit of the CD dynamic range worse than the maximum signal strength of vinyl and even if you consider quantisation noise worse than real noise, CD is still not worse than vinyl.
What about human hearing? Here the dynamic range is about 120 dB, more than can be had from either CD or vinyl. But even if a medium could support 120 dB of dynamic range, it would be totally impractical for any real world listening.A realistic dynamic range of a symphony orchestra would be 60 dB, but commercial recordings are never mastered to get close to it.
Mastering
There are a lot of CDs that have extreme compression and therefore sound really bad. This is by no means an inherent limitation of the medium, but a priority set by recording engineers. They want to win the loudness wars and prioritise this over sound quality. Compressed recordings are perceived as louder and therefore they stand out when played on the radio.
Vinyl records are usually not compressed this way, therefore they sound better than CDs that are. And there are also a lot of vinyl records around that sound a lot worse than they could have been when they were properly mastered.
It is also rather hard to find vinyl records and CDs that have the identical recording and mixing on them, so you can really compare the sound quality of each medium. This is also true for normal CDs on one hand and Super Audio CD on the other hand. They are not the same mixing to begin with, they sound different and listeners prefer one over the other.
Tweaking
You have people who enjoy music, even if it is heard on less than ideal equipment, and you have audiophiles, who constantly tweak their equipment and buy super expensive cables on a never ending quest to improve sound quality. Record players have more potential to be tweaked than CD players, therefore they tend to be liked more by audiophiles.
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