Different axes take less time to complete certain tasks. The woodsman's axe, for example, is the best axe at cutting down trees. The broad axe is the best at forming tree trunks into logs and smoothing log cabin walls. The carving axe is the best at carving and carving log cabin walls. The splitting axe is the best at splitting logs and tree trunks into blocks and splitting blocks into firewood. In general, paying attention to the message given when attempting an action will reveal the best tool for the task.
If you choose Lonely Settler starting scenario, you will acquire several axes.