Is there any difference between these two languages? If I have the following two languages:

$L_1= \{ w \in \{a,b\} |$ $w$ has neither $ab$ nor $ba$ as a subword$\}$

$L_2= \{ w \in \{a,b\}^* |$ $w$ has neither $ab$ nor $ba$ as a subword$\}$.

Is there any difference between $L_1$ and $L_2$? I understand the impact of the Kleene star, but I don't see what difference it makes here.
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, $L_1 = \{a, b\}$.