What user29999 said was the main difference, i.e.: a distance is a function
$d:X \times X \longrightarrow \mathbb{R}_+$
while a norm is a function:
$\| \cdot \| X \longrightarrow \mathbb{R}_+$
However, I think that you wonder whether once induces the other. So a norm always induces a distance by:
$d(x,y) = \|x-y\|$
However, the other way around is not always true. For a distance to come from a norm, it needs to verifiy:
$d(\alpha x, \alpha y) = |\alpha | d(x,y)$
If we take the discrete distance on any space:
$d(x,y) = \begin{cases} 1, \text{ if x = y}\\ 0, \text{ if x $\neq$ y} \end{cases}$
Then this distance does not verify the condition, e.g. for $\alpha = 2$.