Let $E$ be the subspace of $\mathbb{R}^{n+1}$ for which the coordinates sum to $0$ and let $\Phi$ be the set of vectors in $E$ of length $\sqrt{2}$ and which are integers vectors.
It is known that $\Phi$ is a root system of finite type $A_n$. One choice of simple roots expressed in the standard basis is: $\alpha_i=e_i-e_{i+1}$, for $1\leq i \leq n$. The reflection w.r.t. $\alpha_i$ permutes the subscripts $i, i+1$ and leaves all other subscripts fixed. Thus $\sigma_{\alpha_i}$ corresponds to the transposition $(i,i+1)$ in the symmetric group $S_n$. These transpositions generate $S_{n+1}$, so we obtain a natural isomorphism of the Weyl group $W_{\Phi}$ corresponding to the root system $\Phi$ onto $S_{n+1}$.
The first question is the following:
In order to compute all the elements of $W_{\Phi}$, can I just express each element of $S_{n+1}$ in terms of the adjacent transpositions and then use the bijection to map it to the element (reduced word in simple reflections) in $W_{\Phi}$?
Also, in wiki entry on Weyl groups I read that the Weyl group for $A_n$ is the permutation group on $n+1$.
Probably a silly question, but why don't they just say symmetric group instead of permutation group? What is the distinction here?
First question: Yes.
Second question: I've heard both names used. I guess by "permutation group" people might more generally mean a subgroup of some $S_m$ in some contexts, but in this case, it's definitely the "full" permutation group / symmetric group $S_{n+1}$. (By the way I think your term "symmetry group" is also non-standard, as it refers to the group of all symmetries of a geometric object. Incidentally, the symmetry group of the root system $A_{n}$ is larger than its Weyl group for $ n\ge 2$.)