DOF or Degrees of freedom refers to the number of directions a body, or in our case a simulator platform can move. This has nothing to do with the travel distance of a certain actuator or the angle a rotary drive can achieve. It refers to the different specific "motions" the simulator can make or simulate. The maximum is alway 6.
An Axis is what makes a simulator move, in our case this can be an electric servo driven linear actuator or a servo driven rotary actuator. Simply put, an axis is an electric, hydraulic or pneumatic "motor" that makes the simulator move. You can have more axis/"motors" per degree of freedom, but never more degrees of freedom than you have "motors" on your system. So if you want a 3dof motion simulator, you need at least 3 different axis. But 4 is fine to. It is more expensive, but also stronger or more rigid. An extra axis can be used if you want to add a certain dimension to a dof, for example use a "g-seat" on a motion platfom. The way the actuators are orientated determines which degrees of freedom you can use.
Take the F-15 in the picture above, it can move forwards or backwards. Moving in this direction is what we call translating along the longitudinal axis or: Surge.
Going from left to right in this case is translating along the lateral axis or: Sway.
Moving up and down, you are translating along the vertical axis or: Heave.
Banking left or right, you are rotating along the longitudinal axis or: Roll.
Banking forwards or backwards, you are rotating along the lateral axis or: Pitch.
Turning around where you are standing, you are rotating along the vertical axis or: Yaw.
These are the 6 degrees of freedom and combining these will allow you to make any type of motion.
Another example is your head. It has 3 dof in regards to your body . You can look up and down (Pitch), left and right (Yaw) and tilt your head left and right (Roll), but it cannot go anywhere on its own (no Surge, Sway or Heave).