What is the difference between a permutation group and a symmetric group?
I know such questions asked before on Math Stack Exchange, for example here.
But I would appreciate it if someone could spell out the difference without discussing group actions.
A symmetric group on a set $X$ is the set of all bijections on $X$ (called permutations) under the operation of function composition.
A permutation group is a subgroup of a symmetric group. Be careful, however, because Cayley's Theorem states that every group is isomorphic to a subgroup of a symmetric group; so to be clear: a permutation group is a group of permutations, not necessarily equal to a symmetric group.
For example, $G=(\{e, (13)\}, \circ)$ is a permutation group, since it is a subgroup of the symmetric group $S_3$. But $H=(\{1,-1\},\times)\cong G$, despite $H$ not being a permutation group.