Of course, this is subject to interpretation, and it may be regional, but here is what first pops into my mind when I hear those two words:
Napkins are used to, say, clean barbecue sauce off my fingers or cheeks; tissues are used to blow my nose (although, in a pinch, I could use one in place of the other). Tissues are generally made gentler, so as not to irritate the sensitive skin around the nose, while napkins are made sturdier, so they don't fall apart when wiping something sticky. That said, it would hardly surprise me of other countries or cultures used the words differently. After all, a biscuit in Alabama is not the same thing as a biscuit in Wales.
As a footnote, there are other ways these words can be used, where they would mean something totally different (e.g., sanitary napkins, or toilet tissue). Other related items include paper towels, and towellettes (or "wet naps").