I think by saying this, Jesus was refuting claims that people shouldn't work because the Messiah had returned. Possibly he didn't want accusations that becoming a Christian made you lazy and of no use to the present world. Moreover, some slaves had thought that Jesus was a literal liberator - removing all wordly authority so slaves no longer had to work for their masters. He wanted people to be a good witness of God - by impressing their masters they would also prove a positive testimony to the Lord.
The difference, I believe, is in the motivation. In general, Carpe Diem, usually means make good use of opportunities while you may - you might lose the chance. The Christian equivalent would be Ecclesiastes 9:10 Whatsoever thy hand findeth to do, do it with thy might;
Your first quotation, however, means generally that God is your judge, not man, and your service in whatever sphere should be for Him.