Specifications: Never buy a refurbished desktop PC without learning more about the PC’s overall specifications, like the RAM, GPU, Motherboard, etc.
Warranty: Warranty should be your primary concern when you are buying a refurbished desktop PC. A shoddy warranty deal should be enough to let you walk away.
The aesthetic part of the whole deal would not be the least of your concerns. You do not want something that is falling off the seams.
Price: A refurbished product is not a brand-new product. As a result, you need to adjust the overall price point. There is no point in getting an expensive, refurbished PC at the price of a new PC.
Needs: As a desktop enthusiast, you need to have a clear idea about the kind of work you will be doing with your PC. As a result, adjust your expectations.