This is not an answer in algebra, but I wrote the following without noticing that you had asked for examples in algebra. So here it is.

I like the Mayer-Vietoris spectral sequence for giving easy examples. For simplicity, suppose that $K$ is a simplicial complex expressed as a union of $n$ subcomplexes $K_1,\ldots,K_n$. You can make a long exact sequence of chain complexes, with the degree zero bit being the chain complex $C_*(K)$, the degree one bit being the direct sum $\bigoplus_i C_*(K_i)$, the degree two bit begin the direct sum $\bigoplus_{i<j}C_*(K_i\cap K_j)$, the degree three bit being $\bigoplus_{i<j<k}C_*(K_i\cap K_j\cap K_k)$, etc.

For one of the two spectral sequences arising from this double chain complex, the $E^1$-page is all zero. For the other one, the $E^1$-page is the homologies of $K$ and all of the intersections of the pieces. If you only have two pieces, the final differential is $d^2$, which carries the same information as the connecting homomorphism in the standard Mayer-Vietoris long exact sequence.

How can the spectral sequence be helpful? I was once shown a `proof' that $H_m(K)=0$ for some $K$ made from $n$ pieces in this way, because $H_{m}(K_i)=0$ for all $i$ and $H_{m-1}$ of each intersection was 0. Knowing about the spectral sequence makes it easy to see that there is a mistake in this argument: what you really need is that $H_{m-k}=0$ for each of the $k$-fold intersections for each $k$.

For an example where a higher differential is non-zero, consider this spectral sequence for the reduced homology of the $(n-2)$-sphere, made as the boundary of an $(n-1)$-simplex. This triangulation of the $(n-2)$-sphere consists of $n$ top dimensional simplices $K_1,\ldots,K_n$ and each intersection of some proper subset of these is a simplex, while the intersection of all $n$ of them is empty. In the $E^1$-page of the non-trivial spectral sequence, every entry is zero except the zero column, which contains the reduced homology of $S^{n-2}$ and the $n$th column which contains the reduced homology of the empty set (i.e., $\mathbb{Z}$ in dimension $-1$
and 0 elsewhere). The differential $d^n$ must therefore be an isomorphism, showing that the reduced homology of $S^{n-2}$ is zero except in degree $n-2$ where it is $\mathbb{Z}$.

functorialityof formation of spectral seq. the isom problem reduces to case of sheaf-Ext's. That's awesome: a problem for global invariants reduces to sheaf version, which is "local". Usedall the time(etale cohom, duality,...) to prove isoms. $\endgroup$