Two finite non-isomorphic groups with the same order profile: let $C_n$ be the cyclic group of $n$ elements, let $Q=\langle\thinspace a,b\thinspace|\thinspace a^4=1,a^2=b^2,ab=ba^3\thinspace\rangle$ be the quaternion group, then $C_4\times C_4$ and $C_2\times Q$ are not isomorphic (the first is abelian, the second is not) but both have 1 element of order 1, 3 elements of order 2, and 12 elements of order 4.
By contrast, if two finite abelian groups have the same order profile, then they are isomorphic.
