A Spiral starts at a centre point and flows away from said point in a circular path with a progressively increasing radius on a flat plane so is two dimentional.
A helix starts at a point on a circumference and rotates around the circumference at the same radius, if it is a parallel helix, moving forward with a given pitch eg. a screw thread. It can also move out with a progressive radius as on a cone. With the cone, when viewed from the tip, you can see a spiral in two dimentions but you won't see the helix as the always have effectively some form of forward motion.
Also, just in case someone notes, a helix can also have a variable pitch but I don't know why you would want one.
Spiral staircases are so called because of how they look from the plan view where the tread appears to rotate out around a given centre point ie the column. which geometrically is actually a cylinder with a very very deep thread with an equally deep pitch wrapped around it therefore a helix.
Edited By Bodgit Fixit and Run on 31/01/2015 15:48:43