The Climb Limited TOM is controlled by the requirement to make a still air gradient - hence wind cannot affect it. The Obstacle Limited TOM is controlled by a requirement to clear obstacles by a given margin, and this is affected by wind. Headwinds improve the aircraft's gradient relative to the ground (and thus increase the Obstacle Limited Mass) and tailwinds degrade it.
The Climb Limited TOM used to be called the WAT Limit, because only Weight, Altitude, and Temperature affect the still air climb gradient. I've always thought that a much less confusing name.