And adze is similar to an axe, but with the blade rotated 90 degrees. Like an axe, you use it to chip out chunks of wood. But the adze works more like a large wood plane. You use it to dig into the wood, and pop out chunks of wood. Like using one with a curved blade to "hollow" out things - like making a large spoon, a large wood bowl, or even a dugout canoe. The adze with a straight blade was used to finish squaring up a log. The Broadaxe to the main job of getting the log into a roughly square shape, and the adze then was used to "smooth" up the rough sides. It would leave a slightly wavy look on the log.
The other way to think about it would be to compare the axe and adze to the whittling knife and crook knife. The regular whittling knife cuts mostly straight/flat, while the crook knife is used to dig into the wood to hollow out certain areas.
I hope this helps answer your questions.
Mikey - that grumpy ol' German blacksmith out in the Hinterlands
p.s. I've made a few small "bowl" adzes over the years. They are being used by several friends to chop out large bowls and troughs from logs.