I have gotten in the habit of using the stock browser due to a perseption that it was faster and I liked the usability a bit more, BUT, after getting my S4 I needed to access my banks website to check my accounts and the stock browser repeatedly failed to get past the online banking security! After several calls to the bank we gave up and I did my business over the phone! I tried again several times over the next 3 days and no go, finally on a whim I popped up Chrome and it took me right through the verification screens and accessed my account with no problem! Interestingly enough, the results had been just opposite on my S3!?
Side note...The tech in the IT department at the bank speculated that it may be the new security software implimented in the S4, but he admitted that was a guess!
Cheers