Does the S4 have a SIM card and regular phone service? If yes, and you are using your carrier number through your carrier's regular texting feature, then no, the texts will not sync.
The way to fix this is to replace your carrier's standard texting feature with an internet data based texting app, like Google Voice or Whatsapp or others you can find in the play store (some of which assign you a new phone number).
If your carrier is Verizon then you can use the Verizon Messages app works great I am using it on my nexus 7 2013 even with the phone turned off you still receive your sms & mms on the tablet. Hope this helps