As verbs, there is a slight difference. "Affix" means you stick something to something else. "Affix stamp here", written on on envelope, means "stick the stamp on this spot". "Append" means to add something to the end. Usually this means adding something to a written document (certificates, extra notes, an appendix, and so on). I'd expect to see them in business or academic English, or at least in a set of instructions.