Here is the solution:
function findDifference(nums1, nums2)
set1 := convert nums1 to set
set2 := convert nums2 to set
answer1 := empty list
answer2 := empty list
for each element in set1
if element not in set2 then
add element to answer1
for each element in set2
if element not in set1 then
add element to answer2
return [answer1, answer2]
You compute both differences separately. Time: $O(n + m)$, Space: $O(n + m)$.