How would the chart around a point vary be different to the tangent space at the same point?

Question

If I'm not mistaken, the tangent space and the space that a chart maps to are both Euclidean. So if I take some points from a neighbourhood around a point on the manifold, and map them to Eulclidean space with a chart; how might they be different to the collection of tangent vectors I would get were I to use the exponential map to map them to the points tangent space?