One fundamental difference concerns the behavior of Postnikov towers, or the relationship between the spectrum/simplicial abelian group and its homotopy groups. In simplicial abelian groups all Postnikov towers are splittable, since there are no higher Ext's between abelian groups; thus every simplicial abelian group is equivalent to a product of K(A,n)'s. But in spectra there are lots of higher Exts between Eilenberg-Maclane spectra (like those corresponding to Steenrod powers), and this means that for a general spectrum its homotopy groups will have complicated relationships with each other, encoded for instance in
the spectrum's "k-invariants" (the most basic instance of this, by the way, is that the homotopy groups of a spectrum are graded modules over the stable homotopy groups of spheres, whereas there's no extra structure for simplicial abelian groups).
A related perspective would be that the richness of spectra vs. simplicial abelian groups corresponds to the richness of the Steenrod algebra (acting on cohomology of spectra) vs. just its Bockstein part (which is all that acts on "cohomology" of simplicial abelian groups).
But maybe the most compelling picture illustrating the differences is the chromatic one. In some sense the chromatic picture of stable homotopy theory tells you that the difference between spectra and simplicial abelain groups lies in the existence of fundamental and systematic periodic phenomena in the former which are completely lacking in the latter. Concretely, many connective spectra are harmonic (i.e. completely amenable to chromatic techniques, i.e. canonically filterable with graded pieces displaying controlled periodicity) -- for instance, finite spectra (chromatic convergence theorem) and suspension spectra (result of Hopkins and Ravenel) -- whereas simplicial abelian groups are only sensitive to the 0th chromatic layer, i.e. rationalization (so nothing periodic about it at all), since their higher Morava K-theories vanish, by the splitting of the Postnikov tower alluded to above and the fact that you can't make a bounded above spectrum periodic without killing it.