The stock email client only polls for mail which means it sends a request to the server to check for mail at the stipulated time intervals, on the other hand the Gmail app pushes for mail which is what your referring to as sync and receives mail as soon as they are received by the server.
I'm not sure what would cause this, as I said above I do not have this problems with any of my four email account all polled through the stock email client.
Have you double checked your account settings? Or as Prime suggested above you could try an alternative email client, there several on Google Play. I did use K9 on my previous Android phone because the stock client was so terrible but I actually like the vanilla Android email app.
Hope something there helps.