Here is my take on the difference.
For a short answer, "sticking" means staying in contact with the opponent so that he cannot separate to execute an effective technique. "Adhering" means making the opponent stay in contact with you so that you can uproot him and make him stay in contact with your technique.
For a long answer, consider Sunzi's (Sun-Tzu's) Art of War
, especially Chapter 5, Lines 21-23 and Chapter 6, Lines 7, 17, 22-24, 26-34. As we have discussed before on this forum, understand that the title of Chapter 5 (translated on my hyperlink as "energy") is usually translated in Tai Chi contexts as "posture" (e.g., the 13 Postures) or as "tendency" (e.g., going along with the opponent's "tendency) and the title of Chapter 6 is usually translated as "Empty and Full" or "Substantial and Insubstantial."
Also consider the Daodejing (Tao Te Ching)
, expecially chapter 78.
The strategy of Taijiquan is to use the soft to overcome the hard--or perhaps more clearly translated--to use the yielding to overcome the unyielding. If, however, Taijiquan truly used only yielding, this would be a violation of the philosophical doctrine of Taiji, which says that yin and yang can never separate. In Taijiquan, using the yielding as a strategy must therefore have its own yin and yang.
In my current view, the yin part of Tai Chi "yielding" is what we generally call "following" and the yang part is what we generally call "sticking." With these two in harmony, you can both frustrate and control the opponent through his own actions (or inaction).
"Yielding" also has its yin and yang. I would think that the yin of "yielding" is what is called "following" ("sui2") as a term of art, and the yang part is what is called "connecting" ("lian2"). To properly "yield," you have not only to avoid resisting the opponent, you must also avoid losing contact so that there is something to yield to.
"Sticking" also has its yin and yang. What is properly called "sticking" ("nian2") is the yin part. This involves putting some of yourself into every move of the opponent, so that you can always add the 10 percent in the wrong place that frustrates his purposes. What is called "adhering" is arguably the Yang part. Where the opponent seeks pressure, you gladly offer it, but perhaps 10 percent less then what he needs. This way, the opponent always tries for more, and you can make him follow you to his disadvantage.
If you stick to the opponent and make the opponent stick to you, you always have leverage to affect his empty and full. You have him coming and going.
The typical fault in trying to "follow" is that you do too much and get "ahead" of your opponent. You do not let him go where he is trying to go, and you resist. In resisting, you deny yourself the energy you need to use against him and provide a handle he can instead use against you.
The typical fault in trying to stay "connected" is that you do too little and you lose your way. If you have lost track of your opponent's tendency, how can you continue to follow? How can you use the interplay of yin and yang, empty and full if there is a link missing from the chain of cause and effect?
The typical fault in trying to "adhere" is that you do too much and go against your opponent's motion. Your opponent will not try to go where the opposition appears too fierce. You must give pressure where the opponent wants (or perhaps needs) pressure in order to encourage the opponent to move. Too much pressure, and you signal the opponent to stop or to choose another path.
The typical fault in trying to "stick" is to do too little and not fill up the "empty space" between you and the opponent. Your "flatness" opens up gaps between the opponent that allows him freedom of action.
I believe that Yang Chengfu said that distinguishing empty and full is the number one rule in Taijiquan. If you ponder the strategic, tactical, and physical ramifications of this, you can get some ideas for how you might want to train push hands.