There are several different "technologies" at work, and it can be pretty tough to figure out just what's going on.
"Commercial" CDs are made with processes not available to most of us, and can have the information "molded in" or "stamped on," or can be "burned" at power levels that OSHA won't let us po' folk use. These are "done" when you get them, and can't be changed.
Audio CDs (CD-R) that you write on your PC use a "simplified" file system, that only allows a single "address block." You write the "sounds" to the disk, and then your burner program creates an index to tell a reader where to go (on the disk) to get the songs and play them in order. The sound information is useless until the "index" is written, and you cannot "re-index," so it's a write-once and you're done process.
If you use "audio cd" blanks, there are pre-recorded files that let "music box" burners use them, and you also pay a "pro-rata" royalty fee, included in the purchase - on the assumption that you'll be stealing somebody's music, so you pay a little extra up front. This makes the "audio" blanks somewhat more expensive, but does not give you a license to copy a specific title. "Data CD" blanks can be used to write music to CDs in your computer, are cheaper, and (if you're so inclined) you can go pay the copyright cops, if needed.
You can use a CD-R for computer data files in an "open" state. You write the data files, and the burner makes an "index;" but you can add a new "index" later if you didn't "close" the burn. You can NOT erase anything, but if you add new data files, a new "index" is created, that shows where all of the old files - and the new ones - are. The old index remains there, using a little disk space, but only the last index written can be accessed. Any files you "delete" remain on the disk, but can NEVER be re-accessed, since they're not in the current index. They are not "erased," although they are "gone" because you can't ever open them.
You can "close" (some call it "finalize") a data CD-R, and if you do, it's just like an Audio CD-R. You're done, and can't change or add anything to it. There would seldom be a reason to do this on a data CD, unless you are trading CDs with someone who has a different setup. A few machines have trouble reading "foreign" CDs that haven't been "finaled." With most burner software you have to look for the setting and "do it on purpose," and you seldom need to.
The CD-R writes, essentially, by "exposing" a dye layer that is sensitive to the laser light in the "burner." "Burning" is really not a good term, since all you're really doing is "exposing" a photosensitive layer. Once it's exposed, it can't be "unexposed," hence the write-once characteristic. Sort of like taking a photo with conventional film.
A CD-R/W, or "rewritable" disk uses a more specialized dye layer that can be switched back and forth between two "conditions," so you can, within limits, erase and rewrite to it. Some burner software lets you set up your CD burner with a CD-R/W disk in it as an "accessory hard drive" so that you can "drag-and-drop" stuff on and off of it. I don't know that anyone I've run into uses them this way, but it could be useful for "storage." The process is too slow to be practical for active files (on my machines) and external (USB2) hard drives are too cheap to justify messing with it.
Others may disagree, but I have a hard time justifying the price difference to use CD-R/Ws for anything, and especially for any audio. I don't know what would be the effect of "closing" so that most audio playback devices could read them, but this normally "write protects" the disk, so you would still not be able to change them later, although you might be able to reformat (erase) them for reuse(?). I'd prefer to copy the old CD version to HD (hard drive), do my recomposing, and burn to a new blank CD-R. Keep or discard the old CD at your own option - they're really pretty cheap now.
The newest "bug in the skillet" is the appearance of DVD-R/RW "burners." A DVD-RW drive can burn DVD blanks, and if it carries the "Compact Disk ReWritable" logo, can burn CD-R or CD-R/W disks. Be aware that it's likely that a "DVD-R/RW (CD-R/W)" burner will frequently be much slower at burning ordinary CD-R disks than would be implied by the "sales specs." A dedicated CD-R/W is probably still a better choice for most.