In CentOS, what is the difference between yum update and yum upgrade?

yum upgrade forces the removal of obsolete packages, while yum update may or may not also do this. The removal of obsolete packages can be risky, as it may remove packages that you use.

This makes yum update the safer option.

From man yum:

update

If run without any packages, update will update every currently installed package. If one or more packages or package globs are specified, Yum will only update the listed packages. While updating packages, yum will ensure that all dependencies are satisfied. (See Specifying package names for more information) If the packages or globs specified match to packages which are not currently installed then update will not install them. update operates on groups, files, provides and filelists just like the "install" command. If the main obsoletes configure option is true (default) or the --obsoletes flag is present yum will include package obsoletes in its calculations - this makes it better for distro-version changes, for example: upgrading from somelinux 8.0 to somelinux 9.

upgrade

Is the same as the update command with the --obsoletes flag set. See update for more details.

yum upgrade and yum update will perform the same function that update to the latest current version of package.

But the difference is Upgrade will delete obsolete packages, while update will preserve them.