To re-use the same addresses with the new server, it will be sufficient to copy just the wallet.dat. That single file contains the private keys to all receiving addresses that have been used since the wallet.dat file was originally created.
I would recommend copying across the wallet.dat, then making sure your addresses are there and available in bitcoind on the new server. If they are, then the wallet was copied over successfully and you're good to go.
As @Jannes mentioned, you will need to turn off bitcoind on both servers before you do the copy.
1Turn OFF bitcoind on both servers before doing the copying. Don't delete the file from the old server until you're sure it works on the new server. – Jannes – 2015-12-18T23:58:39.487