I said this would open up a can of worms and it has.
To do an A/B comparison between vinyl and CD is virtually impossible for the reasons Adrian has mentioned. The number of variables that arise with the pressing of Vinyl are many: for example how many copies are pressed from a stamper; early pressings will be better than later pressings. Then there is the quality of the vinyl itself. Some of you may remember the oil crisis of the late '60s when LPs used less vinyl than before and the records were much thinner. PVC was sometimes recycled, ether from the trimmings after pressing, or even from scrapped LPs that had been chipped up. The trouble is that PVC is a thermosetting plastic; once set it will not soften and flow on reheating. This means that if used PVC scrap is added to new, the overall mix is inferior to that of virgin vinyl.
There are other factors involved: the pressure, temperature and time in the mould are all important. Get them wrong and a perfectly good looking sample from the press may have internal stresses which result in the record warping in time; something with which very little can be done afterwards.
So that is the manufacturing side of things. Now there is the playback. The idea of dragging a tiny piece of diamond through a wiggly goove of the record is on the face of it absurd - but it works sufficiently well to be practicable. How this moving diamond is supported and allowed to track the record is the next problem. The great majority of tonearms (a misnomer from the days of acoustic gramophones, where the arm was the first part of a loudspeaker horn) are pivoted. It is impossible for a pivoted arm to exactly match the motion of the lacquer cutter: the arm moves in an arc, the cutter radially. Fortunately the acoustic distortion from these geometrical errors are second harmonic, so benign and euphonic. Even radial tracking arms are not without their problems - one of which is the requirement to use as short an arm as possible ( usually at least 6"), so as to prevent any yawing motion as the arm moves. A short arm means any warp wow is exacerbated.
Finally there is the cartridge. Regardless of the type of electro-mechanical generator used, the shape of the stylus and material used for the cantilever are important. The stylus profile has to approximate the triangular cross section of the cutter and the cantilever has to be as stiff and rigid as possible. The grinding and polishing of ever exotic stylus profiles and the machining of exotic materials (boron, and even sapphire or diamond) for the cantilever is difficult and hence expensive. Again the unwanted deviation from linearity are predominantly second harmonic (several percent in magnitude), so are again audibly benign and euphonic.
After all that it is remarkable that vinyl replay sounds as good as it does. IMO it can sound good, very good, but again IMO it cannot better CD for absolute fidelity. Furthermore good vinyl playback requires relatively expensive equipment, the records are fragile and need to be handled carefully and kept clean, whereas CD replay can use a relatively cheap transport, any expense occurs with the sophistication of the DAC.