Hi, to be honest it's a bit difficult to explain the exact differences between these words from a grammatical point of view as I'm no expert. I can give you some example sentences so that you can observe how they are used.
1. Unhe- Personally I feel unhe can mean them/they. It's used more often when the subjects are being passive or acted upon.
a. Unhe coffee de do : Give them/him/her coffee (if it's a senior or elder you can use uhne instead of the usual usse (him/her) even though it's just one person. It's more polite this way.)
b. Uhne thand lag rahi hai (They are feeling cold )
c. Unhe e-mail bhej do (Send them an email)
2. Unhone- It's when they (the subjects) are being active rather than passive that one uses unhone. It's used more often with the past tense.
a. unhone mujhe bulaya (They called me)
b. Unhone hume khana diya (They gave us food)
c. unhone kaam kahatam kar diya (They have finished the work)