Hatcher's theorem 1.38 states that there is a correspondence between isomorphism classes of path-connected covering spaces $p: \tilde{X} \rightarrow X$ and conjugacy classes of subgroups of $\pi_1(X,x_0)$.
The proof claims that given a covering space the basepoint $\tilde{x_0}$ within $p^{-1}(x_0)$ corresponds to changing $p_*(\pi_1(\tilde{X},\tilde{x_0}))$ to a conjugate subgroup of $\pi_1(X,x_0)$.
What's an example of a case when the conjugate subgroup that it changes to is not equal to the subgroup itself? I'm looking for a specific example of an $X$ and $\tilde{X}$.
There is an example to your question in the Hatcher's book itself. Earlier in section 1.3 there is a table of covering spaces of the wedge sum $S^1\vee S^1$. With $X=S^1\vee S^1$ and $\tilde{X}$ the covering space of example $(3)$ and $(4)$ you have your example.