Sad but true - to move an Audiobook download between devices you have to sync from device to device.
So, if you download on your iPad then sync to your laptop to get it on your laptop and then you can sync to your iPhone. (And make sure sync audiobooks is ticked in the Sync dialog for each device).
At least it works and you are not tied to the original device.
So, you can only download once but you can sync it to other devices.
And beware - it is not stored in the Cloud (yet) - so it makes sense to copy to at least one other device for backup.