The two are very different in definition. Definitions:
A set $S\subseteq \mathbb R^n$ is a convex cone if, for any $x\in S$ and any positive real $\alpha$, the vector $\alpha x$ is also an element of $S$. That is, $S$ is a convex cone if and only if $$\forall x\in S\forall \alpha\in[0,\infty):\alpha x\in S$$
For a set $X$, the convex hull of $X$ is the smallest convex set that contains $X$.
You can see a very big difference in the two definitions in the fact that a convex hull is defined in terms of another set, while "convex cone" is a property that a set can either have or not.
That is, it is possible to take a set, $S$, and ask "is $S$ a convex cone?". This question has a yes or no question, depending on $S$. On the other hand, the question "is $S$ a convex hull" doesn't have a yes or no question. You have to change the question to "Is $S$ a convex hull of $X$" before you can answer it.