The grappler can position/move a grappled creature where they want
The grappler and the grappled creature do not share the same space. For example, the PHB shows that a creature with a reach of 10' would be able to grapple someone even 10' away. A creature 10' away can hardly be considered to be in the same square as you.
The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell. (Player's Handbook, p. 290)
Thus, it is entirely possible to maneuver the grappled creature into an area which the grappler is not in. Specifically, nothing would prevent a grappler from positioning the creature in the area of a spike growth. Thus, the grappled creature would suffer the effects of spike growth (damage and reduced speed). Of course, the reduced speed does not do anything because a grappled creature's speed is already 0, but they will take damage.
Creatures do not suffer effects from areas that they are not in
The grappler's movement would not be penalized from the difficult terrain provided by spike growth because the grappler is not in the area for that effect. The grappler and the grappled creature are not assumed to be linked or the same so there is no reason to assume effects would transfer from one to the other. In the real world, doing such a thing would cost more movement because dragging someone through vines is much harder than dragging them on flat ground. However, D&D is not a very good simulator and makes many simplifications.
So, there is nothing in the rules that would prevent this from working.