Sounds like a bad disk, or at least one with a few bad sectors. Quick formats work by just setting a value in a special part of the drive saying all sections are free for use. Regular formats goes over each sector and sets that while also zeroing out the data. So if the regular format fails with some kind of read error, at a minimum you likely have a dead spot on the drive, which if some crucial bit of the OS gets written to, you're going to have problems galore.
If you can't find any info about the make and model of the drive via software, you'll have to pull the drive out of the computer and read the label. Then see if you can find some kind of diagnostic software on that company's website. Sometimes you can mark a specific part of the disk as bad and it'll cause that area to be ignored, letting you continue to use the drive. Of course generally speaking, once you get one bad spot they'll usually start multiplying in greater and greater frequency until the drive is essentially useless. So make sure to keep some money set aside for a new drive or just a whole new computer given that XP is no longer receiving any kind of security patches.