I learn that RHEL and Centos has the exact same source code. So my question here is if one application built for RHEL 6.4, could we just install and make it running on Centos 6.4 without any update?
And another question is RHEL and Centos will always use the same version number for the same source code, right?
Yes, software built on RHEL 6.4 should work as-is on CentOS 6.4 (assuming you don't depend on one of the relatively few RHEL-specific details, mostly to do with branding).
CentOS, except where necessary, uses upstream RHEL sources unmodified. When they need to modify a package they indicate that in the release notes and in the release field of the RPM in question.