Reading the man pages and running a few straightforward tests, I found out this:
texconfig rehash calls mktexlsr, yet a [directories] argument is not passed on.
mktexlsr accepts a [directories] argument.
texhash calls mktexlsr and passes on a [directories] argument. texhash as a command name is easier to remember than "make TeX ls-R", though.
A number of system folders are skipped because of being write-protected to normal users. So, it is better to use sudo texhash if you can, except when (re)hashing texhash ~/texmf.
texhashormktexlsr(that are exactly the same). – egreg Jun 20 '13 at 09:10