Difference between physicist's vector and mathematician's vector

It is a discrepancy in terminology. So the mathematical term for what physicist use would be a Lorentz(or other symmetry group)-invariant vector field. (a covariant version namely a 1-form can also be constructed). They are geometrical objects deep down. So technically speaking a vector field or a 1-form are vectors at every point of the manifold where they are defined.

The usual case where this happens is in the context of special relativity when one first encounters "4-vectors". These are then mathematical vector fields, that is $\mathbf{x} = x^\mu\partial_\mu$ (physicist just care about the components $x^\mu$). Then if you have a Lorentz transformation $\Lambda$ which you can express as a matrix in components $\Lambda^\mu_{\;\nu}$, transforms our vector $\Lambda \mathbf{x} = \Lambda^\mu_{\;\nu}x^\nu \partial_\mu$. If a certain object doesn't transform like this then physicist say it is not a (Lorentz)-vector. One can check the formal construction of such objects in any book on differential geometry.

Physics are usually interested in the symmetries of a given scenario, and since usually vectors that do not follow the symmetry are not "physical", one refers as vectors to the interesting ones.

EDIT: If you are asking for simpler scenarios, just consider in linear algebra where you can see the same issue at work when you rotate your basis vectors but the "arrow", that is what a physicist kind of has in mind, stays pointing in the same direction after the basis change (although the linear combination coefficients, the component, did change).

Yes, vectors in physics obey the vector space axioms, but they also have specific behaviour under rotations so not all mathematicians' vectors are physicists' vectors..