Refurbished computers come from different sources. A refurbished Mac could just be an old model that Apple didn't manage to sell (every time some new model comes out, Apple will do their post to sell off the previous model, but anything they can't sell will become refurbished). It could be a completely unopened box that someone bought, then changed their mind. Or Dad bought his son a MacBook for his birthday without asking Mum who also bought a MacBook - one will end up being sold as refurbished. So that kind of Mac would actually be brand new.
There will also be models that were sold, broke down in the first week, have been returned and fixed. As with everything in life things can go wrong. There have been people who got refurbished Macs that didn't work or were not in an acceptable state, but that can happen with a brand new Mac as well, and that is what warranty is for.
Any refurbished Mac must be one hundred percent working and complete, and it must be one hundred percent clean. The box is not original, and I think there could be cosmetic damage. On the other hand, you could get more than you paid for, like you could get more RAM or a bigger harddisk for free, if a custom-built Mac had been returned.