Is there any difference between these two languages?
Yes there is. $w \in \{ a, b\}$ means that $w$ may be either $a$ or $b$, while $w \in \{ a, b\}^*$ means that $w$ may be a string over $a$ and $b$ including the empty string $\epsilon$. In particular $L_1$ is a language having only $a$ and $b$, i.e, $L_1 = \{a, b\}$.