Trying to make both weapons equally attractive within the constraints of the damage per attack design parameter either leads to equality or the introduction or additional mechanical elements.
That is to say, we can either make them do the same damage and/or let players re-fluff their weapon however they wish* or add extra rules in attempt to make each weapon a 'meaningful choice' over the other, with the classic boons and trade-offs.
Even then, some classes and setups will still benefit more from one than the other. But then.... we need to be clear of our design objective. Are we trying to address the damage disparity between these two weapons or are we trying to make them equally appealing to all classes? We can't have it all.
*It's not a two-handed sword (a large, slashing, heavy bladed object), it's a two-handed axe (a large, slashing, heavy bladed object).