This is a very common thing people get confused. My mother-in-law was livid at a non Disney hotel when she asked for adjoining rooms and we didn't have the door in between (I loved it myself).
Per the Word Thesaurus, connecting is linked. Adjoining is next-door. One easy way to remember is that you are always linked to the people connected to you but you have to be asked to join your next-door neighbors.
Connecting has the door in between. You have to go outside to access the second room for adjoining.