Yes, North American moose have larger antlers than European moose. The largest antlers being from the Alaska/Yukon variety, which are also the most expensive to hunt. As in other antlered game animals, mature animals have larger antlers.
All moose tags in the continental United States are obtained through drawings, and are very difficult to obtain. I have been applying every year since 1988 in my home state of Montana without drawing a moose tag.
There are many areas in Canada and Alaska with over the counter moose tags, but nonresident aliens are required to have guides, which given the large size of moose, is not a bad thing.