A Helix is:
an object having a three-dimensional shape like that of a wire wound uniformly in a single layer around a cylinder or cone, as in a corkscrew or spiral staircase
A Spiral is:
winding in a continuous and gradually widening (or tightening) curve, either around a central point on a flat plane or about an axis so as to form a cone
The main differences is the radius of the curve, the radius of a helix is constant, the radius of a spiral increases or decreases.