In Animal Crossing: Happy Home Designer, the HHD amiibo cards are used for "calling in" (with the amiibo phone) any villager for which you have the card so that you can design their house. I believe that the Welcome Amiibo cards work the same way in this game.
As for Animal Crossing: New Leaf (Welcome Amiibo), I'm not completely sure, as I have not used my amiibo cards for this game yet. However, what I've heard is that when you talk to Wisp and use the HHD cards, he will let you move them into your town, but when you talk to him with the WA cards, he will bring them to your town in an RV in which you can convince the villager(s) to move to your town and order exclusive, rare items!
I just found out (according to this thread) that scanning WAU RV cards in HHD can freeze your game, so please be careful if you do so! I'm not really sure about how to avoid this freezing as I didn't know about it until now, so I would suggest researching it a bit more before scanning the RV cards into HHD.