How do I repeat a theorem number with the llncs class and vary the text between the two occurrences?

The llncs class uses an approach different from amsthm, therefore the other solution is not applicable.

Add the following lines to your preamble:

newcommandrepeatedtheorem[2]{%
  expandafterletexpandafterrepeatedtheoremtmpcsname#1nameendcsname
  expandafterdefcsname#1nameendcsname{%
    repeatedtheoremtmp ref{#2}%
    globalexpandafterletcsname#1nameendcsnamerepeatedtheoremtmp
  }
}

To repeat the number of a theorem, you have to do the following things.

• Define an unnumbered version of your theorem environment. E.g., if you want to recall/preview a theorem, add the following line to your preamble:

  spnewtheorem*{theorem*}{Theorem}{normalshapebfseries}{itshape}
  

  This will define an environment theorem* that looks like the theorem environment, but without numbers.

• Add a label to the theorem whose number you want to reuse, say label{myAmazingTheorem}.

• At the place where you want to preview/recall the real theorem (with the number given there), use the unnumbered version begin{theorem*}...end{theorem*}.

• Immediately before this unnumbered theorem, tell LaTeX to use the number of the real theorem by adding the line

  repeatedtheorem{theorem*}{myAmazingTheorem}
  

documentclass[envcountsect]{llncs}
spnewtheorem*{theorem*}{Theorem}{normalshapebfseries}{itshape}
newcommandrepeatedtheorem[2]{%
  expandafterletexpandafterrepeatedtheoremtmpcsname#1nameendcsname
  expandafterdefcsname#1nameendcsname{%
    repeatedtheoremtmp ref{#2}%
    globalexpandafterletcsname#1nameendcsnamerepeatedtheoremtmp
  }
}
begin{document}

section{Preview}
repeatedtheorem{theorem*}{myAmazingTheorem}
begin{theorem*}[just a preview]
  My amazing theorem that we will discuss later.
end{theorem*}

begin{theorem*}
  An unnumbered theorem.
end{theorem*}

section{The real thing}
begin{theorem}[The Amazing Theorem]
label{myAmazingTheorem}
A theorem
end{theorem}

end{document}