Using regular dish soap in a dishwasher won't really damage the dishwasher per say.
But the main difference between dishwasher detergent and regular dish detergent is the dishwasher detergent is made so it doesn't create suds in the dishwasher.
The regular dish soap does create suds which can be a problem when used in the dishwasher.
When the regular dish soap is used in the dishwasher is builds up suds which can lead to drainage problems in the dishwasher and may leave soap residue on the dishes.
Filters and the pipes leading from the dishwasher could become clogged and lead to the dishwasher overflowing onto the floor from the dishwasher when it cannot drain.
You will likely have to run the dishwasher through a second or even third wash cycle to get all the suds and soap off the dishes and out of the dishwasher when using regular dish soap.
So it's best to use dishwasher machine detergent in the dishwasher to prevent suds building up.
I have used regular dish soap before a few times in my dishwasher when I need to wash dishes and didn't have anymore dishwasher machine detergent but I wouldn't recommend it.