This question follows from another question Relations between two definitions of Lie algebra.
Exponential maps basically links (though not necessarily one one) a tangent vector to a point at a manifold interpreted as a 'distance'/displacement. With exponential maps we link tangent space (a vector space) to the manifold (made a Lie group).
But why we need to, with exponential maps, make a link between a tangent space and the manifold? i.e. what motivates the concept of exponential maps initially? I guess in physical context it enables us to calculate projection distance from initial projection speed, but in other context I don't know why.