Their effect is similar but they are differently programmed. equation* builds on amsmath macros and is therefore more compatible to other amsmath environments and commands.
For instance, within LaTeX's displaymath environment you cannot use a split environment, even if you loaded amsmath. But within equation* it's possible.
amsmath redefines displaymath such that it uses equation*. That means, after you loaded amsmath, there won't be a difference any more between those two environments. Further, LaTeX's displaymath is fragile, amsmath makes it robust.
To see that, here are two lines of latex.ltx: