Why is the Vitali set not necessarily equal to the interval e.g. [0,1]? I don't seem to get the special properties of Vitali sets which makes them different from the intervals, e.g. [0,1].

The Vitali set can not be an interval. As Mariano Suarez-Alvarez mentioned, a well-known result is that the Vitali set is not measurable, but every interval is measureable. 
However, even more elementary than that: A Vitali set contains exactly one element from each coset of $\mathbb{R} / \mathbb{Q}$ (as additive groups). Alternatively, you can define the equivalence relation on $\mathbb{R}$ by $a \sim b$ if and only if $a - b \in \mathbb{Q}$. The Vitali set is then a set containing exactly one element from each equivalence class. $\mathbb{Q}$ is the equivalence class containing $0$. Hence, Vitali set only contain one rational number. The Vitali set can not be a interval since intervals contain more than a single rational number. 
