(Your first question was answered by i.m. soloveichik in the comments.)
A free group generated by a set of generators is, in a sense, the largest group you can make out of those generators.
A free abelian group generated by a set of generators is, likewise, the largest abelian group you can make with those generators.
Being "free" essentially means the group is not subject to any relations. Elements having finite order is a special case of satisfying the relation $a^n=1$ (or $na=0$, when our group is written additively).
The nonabelian one will usually be larger, because they are not subject to the simplification that commutative operations provide.
I'm not an abelian group person, but I guess that if you take the free group on some generators and mod out the normal subgroup generated by elements of the form $aba^{-1}b^{-1}$, then the quotient group will be the same as the free abelian group on those generators. (Modding out that subgroup forces the quotient to be commutative.)