Could not see the video link, but to be clear, the idea is to scale the geometery (your poly or nurbs model), so it will "follow" along the wire deformer curve. Dont scale the wire curve. the curve is a path the growth will follow. The 3d leather texture, representing cells, does not get scaled either, so the geometery grows thru the texture, and looks like new cells are being added as the vein grows.
Scaling is a simple way of growing the vein along the wire. You could also make clusters to pull along the tip and middle thickness and not the base. A lattice can achieve the same effect, by animating the lattice points to "scale" the geometery in a more controlled manner than a simple scale.
The advantage is that you can get a nice tapper on your vein model first, then only worry about growing it with scale or lattice. Its also nice to be able to change your final grown veins position with the wire deformer curve CVs.