Consider the terminal object $1\in\mathbf{Pos}$.
Then the closure of $1$ in $\mathbf{Pos}$ under all weighted colimits is $\mathbf{Pos}$ itself: If $X\in\mathbf{Pos}$, then $X\cong X\cdot 1$ is the copower of $1$ by $X$.
On the other hand, the closure of $1$ in $\mathbf{Pos}$ undel all conical colimits is the full subcategory spanned by the dicrete posets (which is equivalent to $\mathbf{Set}$); this is because any colimit of discrete posets is still discrete.
Thus, it is not possible to express any $\mathbf{Pos}$-weighted colimit as a conical one (otherwise the two $\mathbf{Pos}$-enriched categories above would be the same).