Why is it bad to pick basis for a vector space?

Solution 1:

You have a mission to find out how strong the wind blows on the pick of the mountain. So you climb to the top, put an aparatus and you get a result that it blows 200km/h in one direction. Then you turn it by 90 degree clockwise and get record 80km/h. You end up with such picture:

You know that tangent space to the surface of our planet is 2 dimentional vector space. So you choose coordinates and write your results in your coordinates [200,80]. You are sure that it is all data you can collect and you go back to give an report. In the base you say that wind blows 200km/h in one direction and 80km/h in other. Report receivers asks you additionally how were you oriented while taking first record. But you no longer remember. Hence your mission fails.

Conclusion

While modeling "real world" situations with vector spaces (in particular $\mathbb{R}^2$ or $\mathbb{R}^3$) you must be aware that in reality there is no canonical/natural choice of coordinates. In our situation by choosing coordinates, you assumed that tangent space has canonical coordinates. Obviously it does not. Just stand on the ground, look around and see if there are any.

Solution 2:

Assume you define a concept about vector spaces by using a basis. Then what tell you that this concept remains relevant with another basis ? Generally when a mathematician uses this method, he must prove that his concept doesn't depend of the chosen basis, which can be not difficult but nonetheless annoying.

Therefore, one generally prefers to have inherent properties which do not depend on the basis or the local coordinates and which are directly defined thanks to the manifold/vector space.