It's not like if performed music is 'converted to digital' when recorded. It's simply a matter of different types of recording technology.
It's not like if sound that's recorded, stored and played back with analog technology is better than sound that's handled digitally either. All recordings mean a detoriation of the live sound. At least in some aspects, like dynamics, analog recordings are more detoriated than digital (the fact that many CD's today has a limited dynamic range is due to radio friendly mixing; not to limitations in the CD format). Also, the frequency range covered by the CD format goes well above what normal people are able to hear -- so even if analog sound, in theory, can superceed this, it sould not have an impact on what comes out of your speakers (even if the recording actually contain them, which is doubtful, and your speakers are able to reproduce them, which most speakers won't).
What's more, the massive dependency on mechanics in reproduction adds a load of interference and distortion to analog players. There's hisses, cracks and pops from the interaction between the needle and the record grooves, and varying distortion caused by dust in the grooves; there's hum from the turntable motor and vibrations from passing trucks or your neighbors walking on the floor, all picked up and amplified by your hi-fi system; there's distortion from the needle being exposed to varying pressure and angles during it's travel from the wide outer to the narrow inner grooves; there's variations in speed, from the motor, or because the record is never completely flat (and according to traditional wisdom, anything that force the record against the platter will only make it worse, because it creates tensions in the record material). You might reduce these problems by buying a very expensive turntable and high quality records, store your records in a controlled environment, and play them in an industry standard clean room -- but you'll never get rid of it all.
I bought my first CD player 20 years ago. After comparing some of my favourite music on LP (played on a mid priced Rega) and CD (played on the cheapest NAD player available back then), I've never turned back. It was muddled and noisy vs clean and detailed. Perhaps I lack the analog fan's ability to filter out mechanical noise from the music. I certainly lack his fondness of muddled (or 'warm') sound.
Of course digital playback is more convenient -- but it's also about getting a real opportunity to get rid of all the problems related to mechanical sound reproduction.