My wife and I have identical GS4s. We both have google accounts, Her default email is a Microsoft @live account. Mine is a Microsoft @msn email account.. Both are configured as POP3 and are set up identically on all screens. Both work fine (receive and send) when on wifi. However, when go off wifi and use the network, I can send and receive without any problems but she cannot send. She gets an authentication error message. I've removed the account and then added it but still have the issue. The only difference between the two is "live" vs "msn." The Verizon store couldn't offer any suggestions. They were actually surprised that the email could be configured as POP3. I have not tried the default setup because I don't want to sync accounts. We use the S4 email for quick looks then delete (without deleting from the server). POP3 allows this, whereas all other modes are active sync. Any suggestions would be appreciated.
I had luck with the POP3 settings for a Hotmail account on my Tbolt, but the MSN Email address refused to sync properly. The account would send on either wifi or cellular data.
I found using the Exchange settings worked best for both on a Galaxy Nexus and Galaxy S4. The MSN account also worked on the Tbolt as exchange. I turned off the contacts and calendar sync for both accounts.
I never got into the habit of deleting emails from my phone. For one account, I do move the emails to a folder on the server if I don't want to see those emails in the main inbox.
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.