Vertically align subfloats at the top while having subcaptions vertically aligned below the subfloats

As suggested by Mico in his answer it can be done with the floatrow package. To make subcaptions it depends on subcaption which depends on caption.

\documentclass{article}

\usepackage{tikz}
\usepackage{caption}
\usepackage{subcaption}
\usepackage{floatrow}
\usepackage{calc}% To calculate width for \FBwidth

\floatsetup{ 
  heightadjust=object,
  valign=t
}

\begin{document}

\begin{figure}
\ffigbox
{%
  \begin{subfloatrow}
    \ffigbox[\FBwidth+0.5cm]% Width of subfloat
    {%
      \begin{tikzpicture}
        \draw circle (1.25cm) {};
      \end{tikzpicture}%
    }
    {%
      \subcaption{Small circle}%
    }
    \ffigbox[\FBwidth+0.5cm]% Width of subfloat
    {%
      \begin{tikzpicture}
        \draw circle (2cm) {};
      \end{tikzpicture}%
    }
    {%
      \subcaption{Big circle}%
    }
  \end{subfloatrow}
}
{%
  \caption{Circles}%
}
\end{figure}

\end{document}

Not an automatic procedure, but not so difficult to use:

\documentclass{article}

\usepackage{tikz}
\usepackage{subfig}

\newcount\bsubfloatcount
\newtoks\bsubfloattoks
\newdimen\bsubfloatht

\makeatletter
\newcommand{\bsubfloat}[2][]{%
  \sbox\z@{#2}%
  \ifdim\bsubfloatht<\ht\z@
    \bsubfloatht=\ht\z@
  \fi
  \advance\bsubfloatcount\@ne
  \@namedef{bsubfloat\romannumeral\bsubfloatcount}{%
    \subfloat[#1]{\vbox to\bsubfloatht{\hbox{#2}\vfill}}}%
}
\newcommand{\resetbsubfloat}{\bsubfloatcount\z@\bsubfloatht=\z@}
\makeatother

\begin{document}

\begin{figure}
\centering
\bsubfloat[Small circle]{%
  \begin{tikzpicture}
    \draw circle (1.25cm) {};
  \end{tikzpicture}%
}
\bsubfloat[Big circle]{%
  \begin{tikzpicture}
    \draw circle (2cm) {};
  \end{tikzpicture}%
}
\bsubfloati\qquad\bsubfloatii
\caption{Circles}
\end{figure}

\end{document}

The \bsubfloat commands are just for doing the calculation, then you can use \bsubfloati, \bsubfloatii, \bsubfloatiii, \bsubfloativ and so on (they are reset on each figure environment) to set up the row of subfloats.

If you have multiple rows, you should use \resetbsubfloat between them and use again \bsubfloati and so on for the placement.

More automatic procedure

Here is one, it has some weaknesses, but it should work.

\documentclass{article}

\usepackage{tikz}
\usepackage{subfig}

\newtoks\bsubfloattoks
\newdimen\bsubfloatht

\makeatletter
\newenvironment{bsubfloatrows}[1][\quad]
  {\def\bsubfloatspace{#1}\resetbsubfloatrows
   \def\\{\printbsubfloatrow\resetbsubfloatrows\par
     \@ifnextchar[{\bsubfloatvspace}{}}%
   \def\bsubfloatvspace[##1]{\vspace{##1}}%
  }
  {\printbsubfloatrow}
\newcommand{\bsubfloat}[2][]{%
  \sbox\z@{#2}%
  \ifdim\bsubfloatht<\ht\z@
    \bsubfloatht=\ht\z@
  \fi
  \bsubfloattoks=\expandafter{\the\bsubfloattoks
    \bsubfloatspace\subfloat[#1]{\vbox to\bsubfloatht{\hbox{#2}\vfill}}}%
}
\newcommand\resetbsubfloatrows{\bsubfloatht\z@\bsubfloattoks={\@gobble}}
\newcommand{\printbsubfloatrow}{\the\bsubfloattoks}
\makeatother



\begin{document}

\begin{figure}
\centering
\begin{bsubfloatrows}[\hfill]
\bsubfloat[Small circle]{%
  \begin{tikzpicture}
    \draw circle (1.25cm) {};
  \end{tikzpicture}%
}
\bsubfloat[Big circle]{%
  \begin{tikzpicture}
    \draw circle (2cm) {};
  \end{tikzpicture}%
}
\\[36pt]
\bsubfloat[Small circle]{%
  \begin{tikzpicture}
    \draw circle (1cm) {};
  \end{tikzpicture}%
}
\bsubfloat[Big circle]{%
  \begin{tikzpicture}
    \draw circle (1.2cm) {};
  \end{tikzpicture}%
}
\end{bsubfloatrows}
\caption{Circles}
\end{figure}

\end{document}

As you see, subfloats are enclosed in an environment, where each row is processed separately as regards the height. The horizontal spacing between two subfloats is given as optional argument to the environment (that's one weakness).

A quick introductory remark: According to the latest edition of the l2tabu document, both the subfigure and subfig packages should no longer be used; instead, one should use the subcaption package (from the same author of the caption package).

The problem you describe might best be solved using the floatrow package, as it provides all kinds of ways of aligning both the contents of side-by-side (sub)figures and their respective captions. Unfortunately, I find the writing style of the package's manual rather hard to penetrate, so I wasn't able to come up with a working MWE. Hopefully you'll have better luck. :-)