The Archives paper is nothing equivalent to the Certificate of Origin. It's just like a copy of the record of the watch, stating the movement no., case no., calibre used and some details like the dial type and color,etc. It also has the date of sale and manufacture. It doesn't tell you anything about the condition of the watch, for they don't need to see your watch before they issue the Archives paper. It's just an extract copy from their record. You can apply to PP via the Internet (just furnish them with the movement no., case no. etc..).
The reason I suggest you ask your seller to apply for it is that if they can't get the Archives paper for the watch, it means there's something wrong with the watch (e.g. it's a stolen watch) and you can cancel the deal. It's just for your own protection. And just for your information, the fee for an Archives paper is about US100.
As for the Cert of Origin, it's the original guarantee paper. It's the one and only issued for the watch. If you lose it, you can only apply for an Archives record. So, the Cert of Origin is very important to some PP collectors.
Hope this helps. Good luck hunting!