I see no vector input in your textures nodes. What texture coordinates are you using?
Generated textures will often default to either Generated or Object type of automatic UV coordinates when omitted, both of which are local coordinate dependent.
That means geometry based animations with local mesh deformations in object space coordinates will change the sahpe of the geometry and hence its apparent UV mapping for a certain vertex, making it look like an animated texture.
You could use Bezier curve automatic UV coordinates, but those will most likely also deform along with the "growth effect" giving either an animated effect similar to what you have now or a non uniform stretching effect (which I am not sure is what you want or not).
Only way I could see this work is using a manually unwrapped mesh object instead, with controlled UV coordinates based on actual vertex, instead of object coordinates.
You could mate a cylinder object, with a quad based geometry, unwrap it into a orthogonal grid, then scale one of its ends to make it a cone. Subdivide it along its height enough times to suit your needs.
Then add a Curve deformation modifier so it deforms along the curve then use scaling along the local Z axis to give the growth effect. That should give a more static texture appearance even though it will still stretch with the growth. Otherwise you will likely also have to animate UV along with shape to compensate it.