You need to find some signature on your data set that allows to identify the points from the first set (A) with those on the second set (B).
An easy way is as follows:
For every element E in A, find the two nearest points (N1, N2) and calculate the angle between N1,E,N2 resulting in three values: the angle and the distances from E to N1 and N2 (ang, d1, d2).
Find 3 points in A with unique tuples (ang, d1, d2).
For every element in B calculate also the distance to its two nearest neighbors and the angle. Find the 3 points matching those selected from A.
Calculating the rotation is just a matter of geometric analysis.
update: you need 3 points to determine the rotation in 3D space. In 2D, two will do.
update 2: as others have commented on other posts, there may be symmetries in A that would stop you for finding the 3 unique triplets for (ang, d1, d2). In that case, for every one of the selected three points in A, you will have to perform a search over all the elements in B matching their triplets until some combination results in a rotation that works for all the elements in A.
