Just like a circle is a special case of an ellipse … I think that a helix could be considered a special case of a spiral.
From Wikipedia (Spiral):
Two major definitions of “spiral” in the American Heritage Dictionary are:
A curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
A three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a helix.
The first definition describes a planar curve, that extends in both of the perpendicular directions within its plane; the groove on one side of a record closely approximates a plane spiral (and it is by the finite width and depth of the groove, but not by the wider spacing between than within tracks, that it falls short of being a perfect example); note that successive loops differ in diameter. In another example, the “center lines” of the arms of a spiral galaxy trace logarithmic spirals.
The second definition includes two kinds of 3-dimensional relatives of spirals:
A conical or volute spring (including the spring used to hold and make contact with the negative terminals of AA or AAA batteries in a battery box), and the vortex that is created when water is draining in a sink is often described as a spiral, or as a conical helix.
Quite explicitly, definition 2 also includes a cylindrical coil spring and a strand of DNA, both of which are quite helical, so that “helix” is a more useful description than “spiral” for each of them; in general, “spiral” is seldom applied if successive “loops” of a curve have the same diameter.
In the side picture, the black curve at the bottom is an Archimedean spiral, while the green curve is a helix. The curve shown in red is a conic helix.