Different proofs in amsmath with different QED symbol?

Here is a direct solution just using the AMS packages. We set up a new environment myproof, which is proof but containing a redefinition of the \qedsymbol.

\documentclass{article}

\usepackage{amsthm,amsmath,amssymb}

\newcommand*{\myproofname}{My proof}
\newenvironment{myproof}[1][\myproofname]{\begin{proof}[#1]\renewcommand*{\qedsymbol}{\(\blacksquare\)}}{\end{proof}}

\begin{document}

\begin{proof}
  Usual proof.
\end{proof}

\begin{myproof}
  New proof style.
\end{myproof}

\begin{proof}[Short]
  Standard proof
  \begin{equation*}
    a = b. \qedhere
  \end{equation*}
\end{proof}

\begin{myproof}[Short new style]
  New proof style
  \begin{equation*}
    a = b. \qedhere
  \end{equation*}
\end{myproof}

\end{document}

I also tend to use ntheorem as Bernard show. Just wanted to add how I'd do this with amsthm:

\documentclass[a4paper]{memoir}
\usepackage{amsthm}

\newenvironment{proof*}[1][\proofname]{
  \renewcommand\qedsymbol{\rule{3mm}{3mm}}
  \begin{proof}[#1]}{\end{proof}}

\begin{document}

\begin{proof}
  sfadf
\end{proof}

\begin{proof}[Proof of ...]
  sfadf
\end{proof}


\begin{proof*}
  asdfsf
\end{proof*}

\begin{proof*}[Proof of ...]
  sfadf
\end{proof*}



\end{document}

This is quite easy with the ntheorem package. Moreover, it manages correctly the placement of end-of-proof symbols when the proof ends in a displayed equation. I added the cleveref package to show how it manages reference types (if the cited theorem were to become a proposition, for instance, its reference would be changed accordingly):

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}

 \usepackage{amsmath}
\usepackage[thmmarks, amsmath, thref]{ntheorem}
\usepackage{cleveref}

\theoremstyle{plain}
\theoremheaderfont{\bfseries}
\theoremseparator{.}
\theorembodyfont{\itshape}
\newtheorem{theorem}{Theorem}

\theoremstyle{nonumberplain}
\theoremheaderfont{\scshape}
\theorembodyfont{\upshape}
\theoremsymbol{\ensuremath{\square}}
\newtheorem{proof}{Proof}

\theoremsymbol{\ensuremath{\blacksquare}}
\newtheorem{blackproof}{Proof}
\begin{document}

\begin{theorem}[Some theorem]\label{thm:some-theorem}
This is an important theorem.
\end{theorem}

\begin{proof}
This is a very important proof.
\end{proof}

\begin{proof}[of \cref{thm:some-theorem}]
This is a very important proof.
\end{proof}

\begin{blackproof}[of \cref{thm:some-theorem}]
This is a very important proof.
\begin{align*}
    a & = b\\ c & = d.
\end{align*}
\end{blackproof}

\end{document}