I'm not comfortable enough with spectral sequences to answer this question, but let me answer an easier version of this question with spectral sequences replaced by long exact sequences.
In algebraic geometry I think all of the long exact sequences you'll run into ultimately come from deriving some functor between abelian categories. In algebraic topology, on the other hand, a very important example of a long exact sequence is the long exact sequence of a fibration, which ought to come from the "derived functor of $\pi_0$" but can't possibly be obtained from the usual abelian category story because in general it involves nonabelian groups.
There are two ways to go from here (that I know of). One is to broaden your notion of derived functor to include nonabelian examples. I'm not comfortable enough with this story to explain it in detail, and in particular I haven't checked that the details work out, but the basic idea is to replace resolutions involving (co)chain complexes with resolutions involving (co)simplicial objects (the Dold-Kan correspondence tells you that the latter reduces to the former in an abelian category, which is some evidence that this is a good idea).
But there is a more directly topological story: the long exact sequence of a fibration reflects a more fundamental underlying structure, namely that of fiber sequences. Start with a pointed continuous map $f : E \to B$ between pointed spaces. Then we can construct the homotopy fiber $F$ of this map, which (at least after fibrant replacement) gives me a fibration
$$F \to E \to B$$
of pointed spaces. Now the curious thing about taking homotopy fibers is that the homotopy fiber of a homotopy fiber, unlike say the kernel of a kernel, is in general nontrivial: if I now take the homotopy fiber of $F \to E$, I find from a standard lemma about homotopy pullbacks (which takes the same form as the corresponding lemma about ordinary pullbacks) that this is the same as the homotopy fiber of the inclusion $\bullet \to B$ of the basepoint into $B$. But this is precisely the based loop space $\Omega B$! Continuing to take homotopy fibers in this way, I get a sequence of fibrations
$$\dots \to \Omega^2 F \to \Omega^2 E \to \Omega^2 B \to \Omega F \to \Omega E \to \Omega B \to F \to E \to B$$
and applying $\pi_0$ to this sequence gets me the long exact sequence of homotopy groups, using the fact that mapping into spaces behaves well with respect to homotopy limits. (There is a dual story about how to get long exact sequences in homology and cohomology from cofiber sequences which involves repeatedly suspending rather than taking loop spaces, and then either tensoring or cotensoring with a spectrum.)
This is in some sense the nonabelian version of the long exact sequence associated to a short exact sequence of chain complexes, and can be run in any setting where you have a sufficiently well-behaved notion of homotopy limit.
(To get from here to, say, the spectral sequence of a homotopy limit I have been told that the idea is to start by defining a suitable filtration of the homotopy limit. After fibrant replacement this gives a tower of fibrations, and I think patching together the long exact sequences of these fibrations in some way should give the spectral sequence. I'm sure an expert can say more here, though. This should the nonabelian version of the spectral sequence associated to a filtration of a chain complex.)