Let $A$ be a subcomplex of CW-complex $X$. The excision axiom for homology implies that $H_i(X, A)\cong H_i(X/A, *)$, and it is widely known that homotopy groups don't have this property. However, they satisfy a significantly weaker Blakers-Massey theorem. One of its consequences is that when $A$ is $n$-connected and the inclusion $A\to X$ is an isomorphism on the first $s$ homotopy groups, $\pi_k(X, A) \cong \pi_k(X/A, *)$ for $k < s + n - 1$.
Thus to devise an example where $\pi_k(X, A) \not\cong \pi_k(X/A, *)$ it's necessary to take large enough $k$, and the calculation of both sides becomes hard. The original paper of Blakers and Massey claims there are simple examples, but I wasn't able to make them up myself. What are some simple examples of the pairs $(X, A)$ and $(X/A, *)$ with different homotopy groups? Preferably with both $A$ and $X$ simply connected.