Let $X$ be the wedge sum of two circles in $\mathbb{R^2}$, connected together at a point labelled $x_0$. Let $Y$ be the union of two disjoint circles and a line segment joining both together.
Are these homeomorphic? Homotopy equivalent?
It looks like they have the same fundamental group, namely the free group on 2 elements. On the other hand it doesn't feel like I can give a homotopy equivalence from $X$ to $Y$, since there would be discontinuity at $x_0$.
More precisely:
Let $S(x,r)$ denote the circle of radius $r$ in $\mathbb{R^2}$ centered at $(x,0) \in \mathbb{R^2}$.
Then $X$ is $S(-1,1) \cup S(1,1)$ and $Y$ is $S(-2,1) \cup [-1,1] \cup S(2,1)$.
They are definitely homotopy equivalent: retract the connecting line down to a point.
They are definitely not homeomorphic. If they were, any homeo would preserve cut-points (points whose removal results in a disconnected space). But the figure 8 has only one cut-point, the circles-with-line has many.