Hey Mathinik.

The Tangent Line Approximation usually refers to a one-dimensional approximation (or a particular approximation with respect to a partial derivative) or with respect to a directional derivative, but not in a multi-dimensional sense.

The local linearization in a multi-dimensional senses uses what is called the Jacobian and this produces a linear approximation for all variables given all the partial derivatives and thus is not just a line but a plane in n-dimensions. This is usually an approximation (unless the object is linear itself which is a special case).

The approximation is the same as the local linearization.

You also have to consider what the word local means: local usually refers to have an approximation that is based around a particular point or region in the same way that you can expand a function with a known taylor series around a particular point rather than around 0. If you know derivative information around a certain point then you can use a linear approximation (keeping only the first order terms) around that point than having to use higher order terms and this is what is done frequently.