The actual work being done here is by the Yoneda lemma. Brown's representability theorem tells you that these spaces represent the cohomology theory, turning natural transformations into morphisms is done by Yoneda.
That said, uniqueness holds.
Of course, one has to be a little bit more careful about what one means by "natural transformation" in this case. Natural transformation of what? If your natural transformation is of the whole cohomology theory in one go (aka a stable operation) then, indeed, you get a morphism of spectra. If your natural transformation acts on one particular level (aka an unstable operation) then you get a morphism of spaces.
Where you do not get uniqueness is if you have a family of unstable operations (aka a family of morphisms of spaces) which look as if they fit together to give a stable operation. You can get "phantom" morphisms, and there's a "$\lim^1$" term that controls this. A nice place to read about all of this is the papers by Boardman and Boardman, Johnson, and Wilson on stable and unstable cohomology operations (Handbook of algebraic topology, also available via Steve Wilson's homepage).