In a linear analysis, the strain is always proportional to the stress.
In an elastic analysis, the strain follows the same function of stress for loading and unloading. More importantly, strains are reversible meaning that upon unloading, the material returns to its original state without residual stress/strains. For example, the stress-strain curve of a hyperelastic material in the elastic range is nonlinear.
Therefore, a linear analysis will also be an elastic analysis but the opposite is not true. An elastic analysis may very well be nonlinear.