"Wry" evolved from a meaning of "to twist". Applied to humor, it refers to humor that is bitterly or disdainfully ironic or amusing; distorted or perverted in meaning; warped, misdirected, or perverse; words that are unsuitable or wrong; scornful and mocking in a humorous way--it covers a lot of territory (see WordReference.com). Here is a link to examples of wry humor: https://www.pinterest.com/karenegeberg/wry-humor/. A few samples:
Inside every older person is a younger person wondering what the hell happened.
I'm sorry I hurt your feelings when I called you stupid. I really thought you already knew.
I swear; if my memory was any worse, I could plan my own surprise party.
"Dry" humor is often called deadpan humor because it is delivered with a straight face and a serious tone, as if it is not intended to be funny. A good discussion of dry humor here: http://simplicable.com/new/dry-humor.
In general, humor is based on a sense of the unexpected, inexplicable, ridiculous and ironic. The deadpan delivery enhances that because the serious tone makes it more-so, and may even lead to a "delayed reaction", where you don't recognize the humor until later. Wry humor can be delivered in a deadpan manner.
A lot of comedy is delivered with facial expressions or acting out to provide a reinforcing image. Some old-time comics would employ a technique like a drum "rimshot" to flag the punchline. Some humor is just funny on a surface level, or incorporates words that sound funny. What makes dry humor unique is that it isn't always obviously funny, especially with a deadpan delivery; you often need to think about it. The humor is entirely within the meaning of the words.
Steven Wright's humor is a great example of dry humor. Samples here: https://www.brainyquote.com/quotes/authors/s/steven_wright.html. A few examples:
I intend to live forever. So far, so good.
Whenever I think of the past, it brings back so many memories.
I think it's wrong that only one company makes the game Monopoly.
If it's a penny for your thoughts and you put in your two cents worth, then someone, somewhere is making a penny.
What's another word for Thesaurus?
I used to work in a fire hydrant factory. You couldn't park anywhere near the place.
Comedian Henny Youngman was famous for one-liners delivered in a deadpan manner. Much of his humor was both wry and dry. For example, his most famous line:
Take my wife ... please.
If you never saw the delivery, "Take my wife" was said as if he was using his wife as an example to set up a situation, as in "Take my wife [for example]." After a brief pause, the "please" turned it into a request.