Battleaxe is two-handed weapon and has considerable one hand penalty. Handaxe is one-handed weapon/tool and doesn't have this problem.
So, in my opinion, handaxes are better to be dual-wielded with a shield for combat, but of course, battleaxes are brutal if you decide to leave the shield and fight two-handed.
Personally, I prefer using handaxes. One axe for all. Handaxes are balanced for combat and all kinds of timberwork, so you need just one handaxe and don't need to haul different heavy axes each for specific works every day. I only use them when I build carbins. For daily life, one handaxe is enough.