The unit simplex is the $n$-dimensional simplex determined by the zero vector and the unit vectors, i.e., $0,e_1, \ldots,e_n\in\mathbf R^n$. It can be expressed as the set of vectors that satisfy $$x\succcurlyeq0,\quad\mathbf 1^\mathrm T x\le1.$$ The probability simplex is the $(n−1)$-dimensional simplex determined by the unit vectors $e_1,\ldots ,e_n\in\mathbf R^n$. It is the set of vectors that satisfy $$x\succcurlyeq 0,\quad \mathbf 1^\mathrm T x=1.$$
I know that a simplex is the set of all convex combinations of some vectors. I can imagine that in two dimensions, the probability simplex is a right triangle with legs $\mathbf e_1$ and $\mathbf e_2$. In three dimensions, it is a right tetrahedron with legs $\mathbf e_1$, $\mathbf e_2$ and $\mathbf e_3$. But what does the unit simplex look like? What difference can the zero vector make?