These phrases have the same meaning, yes. It sounds most natural to use "unhappy about" with verbs and verb phrases, and "unhappy with" with nouns and noun phrases. This is not a strict rule (and "about" can be used more easily with both verbs and nouns).
"About"
I am unhappy about going to the store. (V)
I am unhappy about studying right now. (V)
I am unhappy about the condition of my new car. (N)
"With"
I am unhappy with my trip to the store. (N)
I am unhappy with my schedule right now. (N)
I am unhappy with the condition of my new car. (N)