A bit of a yes and no answer.
Will the copy contain EXACTLY the same data as the original? Probably not in more than most cases.
Will these differences be noticeable? Depends upon the quality of the playback equipment, and how good the listener is at distinguishing fine differences.
I understand it as: When digital data is written to an optical media (DVD/CD), it is not written in so much of a perfect 0 or 1, on or off, yes or no fashion, but is expressed in more an analogue wave.
The pits and land (0's and 1's for arguments sake) do not have perfectly defined boundaries, with perfectly vertical sides. The side is more of a slope, with a bit of a curve in the bottom of the pit, and a curve near the top, where it meets the land.
The reading laser reflection pickup sees these pits and land, and uses a threshold system, whereby it treats a certain deep depth of pit as a 0 and a particular shallow up to land as 1. The slop and most of the curve gets binned, as the reader is only looking for two values 0 or 1, pit or land. The laser reflection pickup receives varying amounts of laser light reflected back, which it sends to a circuit, which switches between 0 and 1 when the output of the pickup reaches/falls below certain thresholds.
Now, if we take a poor quality burner, and poor quality discs, these pits and the edges are going to be badly shaped, which can introduce errors quite off to what the original data was. Whilst a high quality combination may not be perfectly identical to the original in terms of pit depth and wall shape, it may be seen as no different compared with the original when read with an optical drive.
Of course, when one uses a computer to check the data, software can perform comparisons between the read data multiple times and produce a statistical output indicating the quality of a burn. But there's a big factor and that is the drive one is using in the computer. If the drive has a poor quality laser, poor circuit for determining 0's and 1's or a poor calibration, then who knows if the burn was good or bad?
We have a standalone CD-R (Marantz DR6000) and I use my PC for burning CD-R's, but am keeping the loss less files on the hard disk, encased in archives with error checking. I wonder what the quality of a burn would be like using the digital SPDIF output of the M-Audio 2496 sound card into the DR-6000, compared with a straight 10X burn from the PC's DVD/CD-RW unit - in theory much better, but in practice maybe not?
As for good quality media - both DVD and CD, Taiyo Yuden (Japan) seem to raved about. Verbatim (Japan/Taiwan) also gain praise. There does seem to be a lot of outsourcing by many companies (Verbatim, TDK, Sony, Maxell....) to Moser Baer India to make their optical media. What the quality of these discs are like is anyone's guess. I grabbed some cheap TDK's for the sake of just trailing some music, and they seem reasonable at the moment, but longevity could be awful! There isn't much of a definitive test.
Avoid CD-RW - for longer term storage/ultimate fidelity. These are not burnt in the same way as once-write media. A reaction occurs in the disc which alters the reflectivity (maybe deforming the dye or the metal) and a different system in the read drive is needed to read them. It's harder to read, and arguably harder to write. I wouldn't use them to many times either, as the disc is being deformed each time, and will take the data with less accuracy each time. Great for quick demoing, but not good for ultimate fidelity/quality.