I asked someone to translate "I hope you understand" in Korean and i received two answers.
니가 이해하길 바래. and 니가 이해해줬으면 좋겠어.
is there a difference in intensity of formality? or is it just synonymous words?
The first sentence is definitely more emphatic. It could be translated as
"I want/desire you to understand [and I'm not asking you, I'm telling you]."
The second sentence carries a softer meaning. The 아/어/여 + 주다 (-해줬-) form is a softener that is more akin to saying
"I hope you can understand, please. [And I'm asking for the favor, if you would be so willing please]."