Conservation gives a local symmetry but there may not be a global symmetry.
For instance, you can imagine a physical system with no forces at all, so everything is conserved. But there are still some parameters that define the location of the particles. Then the physical system is locally very symmetric, but it may still have some symmetric global structure where the particles are constrained to lie on a surface of nontrivial topology.
Conservation gives a local symmetry but there may not be a global symmetry.
For instance, you can imagine a physical system with no forces at all, so everything is conserved. But there are still some parameters that define the location of the particles. Then the physical system is locally very symmetric, but it may still have some symmetric global structure where the particles are constrained to lie on a surface of nontrivial topology.