With enough skill an axe can do precision cutting, this is hard on very heavy axes as the head weight is high and the edge is so far in front of the handle. But generally if an axe is light enough to be used for chopping one handed then a lot of cutting can be done with it as well. I would not consider it a replacement for a knife, a small knife is so light it doesn't make a lot of sense not to carry it and have two sharp cutting instruments instead of one. Evolute has also described in some detail how a small knife (sufficiently durable) can be used for axe-like work as well.
-Cliff