What career are you talking about? For example, in the navy, the Officer of the Deck is stationed on the bridge and is in charge of navigation and safety of the ship and is assisted by the Junior Officer of the Deck. Another title could be Assistant Officer.

Asst is short for assistant, often times such as for this case junior would mean the beginning--someone trying to earn up the ranks. Assistant is a solid position usually to aide someone. It really depends on what the title is being applied to, sometimes it's the same rank and sometimes it's not.

I don't think there is any difference. "Junior" implies young in age though. However, if you have an assistant who is older than you, it would not be wise to call him "junior".