wandrinabout,
in still air, they are the same.
But when the air is moving, I think you are forgetting one important fact: All movement has a direction as well as a speed.
Draw the ground speed as an arrow on a paper, with direction and length to represent the speed. Then draw another arrow, starting in the same point, to represent the TAS with the direction of the heading of the aircraft and the length to represent the TAS.
If you draw a third arrow from the tip of the TAS arrow to the tip of the GS arrow, this third arrow will be the wind.
These arrows are what you'd call vectors, and what you just did was in fact to subtract one vector from another. Subtract the GS vector from the TAS vector and you get a third vector representing the wind.
The tailwind component is something else though. It is the part of the wind which blows along the GS vector. The other part which blows perpendicular to the GS vector is the crosswind component.
Probably waaaay oversimplified, but there are probably a few people who are not as knowledgeable as those who post reading this.
Cheers,
Fred