I actually have read both of them in parallel (using a program called ESword).
They are very, very similar. I prefer the KJV for one reason. It was the version my grandpa read the Christmas story out of.
Overall, my personal favorite is the RSV with Apocrypha.
There are a few exceptions. The Lord's Prayer as most of us recite it is actually from Tyndale's translation which was then used in the Book of Common Prayer (nearly all Bibles, including KJV, D-R and RSV use debts and debtors instead of trespasses and trespass against us) and when 1 Corinthians 13 is read at weddings, it is typically from the RSV.