Will that software actually perform better if I use a newer/better
compiler?
This entirely depends on if work is done to the code optimization feature, of the compiler, in a given release.
For instance, right now I use the Mac OS X system version of Clang,
and it's been suggested that I use a later version instead.
You should stick with the same compiler version that you know works, unless you need a feature, that exists only in a later version. Unless you know a later version does something more effective you should not upgrade.
Is that just a suggestion to save me time and memory during
compilation?
The suggestion honestly seems 100% opionion based. If you trust the person that provided this suggestion then you should take it.
I should point out that a compiler, is software, and newer versions can introduce new bugs which result in worst performance in the optimization routinue. You should seek a balance, the best way to do this, is always use the stable release build.