Amalfi Coast or Cinque Terre in June
 

We are going to be on our honeymoon in June and have 5 days in Italy (flying out of Rome on the 6th day) We are planning to go to Paris for 4 days, and Munich for 4 days and looking for a relaxing end to our trip. Which would be better in June? We're looking for some sightseeing but mostly relaxing by the water.

You would be happy with either. Probably the biggest difference to me is that the Cinque Terre had more tourists doing the walking trails, whereas the Amalfi walking trails were much quieter.
If you are not in to walking or trekking then that won't matter. Both areas have good transport from town to town (train or boat in Cinque Terre, or bus or boat in Amalfi/Positano, and both have their own special character!

For a honeymoon however, I don't think you could beat Positano. The beach is nice for relaxing by the water, and its such a romantic place. Get a hotel with a balcony overlooking the water and you've got heaven !
We stayed at HOTEL MONTEMARE, and had views of the sea from our room, it was wonderful, I can recommend it, service was great too.
Good luck

The AC is a long way unless you fly to Naples.
I would let travel logistics dicate.

The two destinations are quite different, even though both are coastal and quite beautiful.

Several years ago I wrote a comparison of these two areas. You might find it useful: http://www.idyllicitaly.com/_files/tip_061027.html

Be advised that because you're flying home from Rome, you will need to spend your last night in Rome. Will you have 4 nights to spend at either destination before continuing on to Rome for your last night?