What is the difference between a symmetry and a change of co-ordinates? What is the difference between a symmetry and a change of co-ordinates in terms of the Lagrangian? The difference physically is obvious: A symmetry relates physically distinct solutions of the equations of motion via the fact that the symmetry manipulation of the fields/particles leaves the action unchanged whereas a co-ordinate change does not physically alter the objects under consideration. But I find it difficult to formulate this mathematically, because a co-ordinate change also leaves the action invariant and maps solutions of the equations of motion (albeit physically the same solutions) onto solutions.  I would greatly appreciate an explicit example demonstrating the difference.
 A: A dynamical symmetry  is a (regular in the appropriate sense) map from the space of states into itself $$s : \Gamma \to \Gamma\:,$$ usually assumed to be injective and surjective,  such that if $$\gamma : \mathbb R \ni t \mapsto \gamma(t) \in \Gamma$$ is a solution of the dynamical equations, $$\gamma':  \mathbb R \ni t \mapsto \gamma'(t) := s(\gamma(t)) \in \Gamma$$
is still a solution of these equations. 
Here there is nothing related to coordinates. $\Gamma$ may be a Hilbert space for instance, if we are discussing quantum dynamical symmetries. 
In case of Lagrangian systems, $\Gamma$ is the space of kinetic states. If a map like $s$ above preserves the Lagrangian, $$L = L\circ s\:,$$ then it is a dynamical symmetry in the sense I pictured above (it is not evident and can be proved!).
The converse is false. Regarding the Hamiltonian formulation replacing $L$ for the Hamiltonian function $H$, where $\Gamma$ is the space of phases, there is a perfect equivalence instead, if you restrict yourself to deal with canonical transformations which do not depend on time and the Hamiltonian function does not depend on time too (there is a more general equivalence but it is a bit technical to explain here).
If $\Gamma$ is a manifold,  coordinate transformations do not act as active transformations on $\Gamma$ (they are the identity map!) thus they are not dynamical symmetries by definition.
However you may sometimes interpret them as active transformations (since $s$ is bijective, regular,  and thus transforms coordinate systems into coordinate systems) referring to suitable classes of coordinates, the ones connected by symmetries! With respect to such a pair of coordinate systems, the studied symmetry in coordinates looks like the identity map.
This coordinate-minded approach is not the right starting point  to understand these notions however, at least for beginners,  in my view.
