A 'quick erase' only creates a new directory, so that your PC (or DVD recorder) sees the disk as empty and then records new data on top of the old files. (i.e. it 'wipes as it goes', as if you were making a video recording, on top of an existing one, onto old-fashioned videotape).
A 'full erase' effectively reformats the disk. It deletes all of the data and blocks out any bad areas which are no longer suitable for recording on.
After a quick erase it's usually possible to retrieve some or all of the data on the disk by using a program such as Isobuster. (However, don't bank on it. The one time that you actually want to retrieve data will probably be the one time that Isobuster can't access it).
It's not normally possible to retrieve data after a full erase although the sort of tools used by forensic scientists might be capable of reading small parts of the data.
It should normally be possible to record just as well onto a 'quick erased' disk as onto a 'full erased' one. However, as stated, a quick erase doesn't block bad areas on a disk. Your recorder should be able to do this 'on the fly' but, if a disk is of borderline quality, you could possibly experience problems. (But, if a disk was of borderline quality it would be best to throw it out anyway).
Chris