I've used both for moderate ice work, sometimes the two together. The axe is a great stopping tool. Though I've never fallen, I certainly practised - releasing myself head first, back over the shoulder, front first etc. You need to know you can do it.
The pick hammer a great tool for pegging and removing nuts, albeit easy to poke your eye when hammering in. The pick hammer is a useful tool for a winter's day emergency if you havn't taken the axe. A treasured tool indeed.