Small space hash functions that are weakly but not strongly universal
Let $H$ be a family of strongly universal hash functions from $U$ to $[m]$.
Construct a new family of hash functions from $U \cup \{x\} \to [m]$ by extending all functions $h \in H$ with $h(x) = 1$. The family is weakly universal since $h(u)$ is distributed uniformly for every $u \in U$, but it is clearly not strongly universal.