|To carry – with an emphasis on the fact that you’re transporting something. If you pick up your suitcase and take it to the airport, that’s 運ぶ。
|To carry a heavy load – unlike 運ぶ、担う emphasizes the heaviness. Can also be used figuratively – as in the English phrase “She bears the burden of caring for her retarded sister.” NP
|”carry specifically on your shoulders” – like a backpack or a rifle.