"Game design" is making the product naturally fun. It involves the intentional emotional design and the psychological phenomenon of flow, which make it inherently interesting and satisfying.
"Gamification" instead uses mechanisms like badges, levels, leaderboards, and points to make users interact with the product.
"Game design" nurtures intrinsic motivation in players, while "Gamification" creates extrinsic motivation for players.
Extrinsic motivation will undermine intrinsic motivation in the long run. This is important not only in product design but in all the fields that require motivation design.
For example, as a teacher, how can we make students be driven by intrinsic motivation more instead of extrinsic motivation? And as a learner, can we first use extrinsic motivation to trick ourselves and develop intrinsic motivation later?