I will eagerly watch for additional discussion of this question, because I believe there are more factors involved than simply whether the sound is stored on a CD or on an LP (vinyl record).
First of all, the original recording signals are most likely different: the LP will record and store an analog signal and the CD (if it's the original recording) will record and store a digital signal.
But someone could take the analog signal from the LP and digitize it for storage on a CD. Would there be a noticeable difference between that sound (originally analog, now digitized) stored on the CD and the sound of that same performance that was recorded originally as an analog signal and stored on the LP? Maybe and maybe not.
Some of the factors that could affect the comparison are: variances caused by the quality of the equipment that was used to make the original recording; which pressing of the LP you are hearing (the first pressing will have a higher-quality sound than the 10,000th pressing); the equipment used for playback (including the stylus on the LP); and maybe even the age of the LP itself.
So, initially, I would absolutely agree with the other posters who say they can hear a difference between a CD and an LP. But if we are talking about a new, first-press LP that was recorded (analog) and played back on exceptional equipment on the one hand, and on the other hand talking about that exact same analog LP recording that has been digitized and stored on a CD - then, I'm not so sure about being able to hear the difference.
But I would love to have the opportunity to find out!