I think the answer to the question could also be; should you use edge or point attack against your opponent, judged by their clothes and armor? The battleaxe delivers a great amount of edge damage, but the point damage from the sword can make it useful against certain types of armor, for example mail, where the sword probably would be better.
You can use shield with both of them, however both the battleaxe and twohanded-sword will get a penalty from not using both hands.