It seems likely that the families are, at least distantly, related. Proving it would be another matter. In theory such arms are inherited, with the main inheritors assuming an un-changed design. Other relatives, possibly cousins, would use a slightly amended version, hence the variation. Spelling changes over time in a similar organic fashion.
The problem is that people can assume that their name comes from the same root as the family that has the coat of arms. Unfortunately, this is not necessarily so. Consequently, people assume the right to use a particular coat of arms when actually their name comes from somewhere else.
The Finnies seem to trace their history back to the Ballyfeeney area. So, if both families can trace their roots to this area, they really could be from the same family. An interseting, if heroic, goal would be to prove or dis-prove the link.