It's a very important choice. You will get both characters, but the first one you get takes over the hotel's lounge. The dragon will turn it into a high-end restaurant, the gryphon will turn it into a nightclub. Pretty much every scene with either character plays out differently depending on who gets chosen first.
There will be two staff members in future builds that only appear depending on which you pick first. Asterion can get an optional uniform later that depends on this choice. Just go with whichever character you like more.