The yellow and dirty trench coats provide a single point bonus to both Charisma and Endurance plus 5 points to Radiation Resistance. The faded trench coat adds 2 points to Perception, however, it has no Radiation Resistance.
Every trench coat variant takes up all of the body slots, preventing armor from being worn over them. They can be upgraded with ballistic weave to increase the protection that they provide.
Locations
Nick Valentine wears a unique faded trench coat that replaces his right hand with a "synth" flesh stripped hand. It can only be removed by replacing it with the faded trench coat given to the Sole Survivor (incidentally "repairing" his hand), allowing his original coat to upgraded with ballistic weave and returned to him.
The appearance of the dirty trench coat is the same as the yellow trench coat on male characters and only shows correctly on female characters. [verified]