How can I save the bounding box of a TikZpicture and use in other TikZpicture

You can use the \useasboundingbox command in the second picture to set the bounding box. Should be the first command in the picture.

\useasboundingbox (0,0) rectangle (<width of first picture>,<height of first picture>);

If you don't know the dimensions of the first picture you can get them from the current bounding box node. Using remember picture you can then access this information in the second node.

The code below will set the accept same bounding box for the second picture.

\documentclass{article}
\usepackage{tikz}
\begin{document}
\fbox{%
\begin{tikzpicture}
   \draw (-1,-1) -- (5,5);
   % more drawing commands ...
   \coordinate (FIRST NE) at (current bounding box.north east);
   \coordinate (FIRST SW) at (current bounding box.south west);
\end{tikzpicture}
}
\fbox{%
\begin{tikzpicture}
   \useasboundingbox (FIRST SW) rectangle (FIRST NE);
   \draw (0,0) -- (1,1);
\end{tikzpicture}
}
\end{document}

This works if both pictures use only positive coordinates. Adjustment must be made if this isn't the case.

The \fbox commands are only to display the bounding box and are not really required.

Another way to do it is to use a single tikzpicture environment and one scope environment:

\begin{tikzpicture}
% TikZ code for first picture
\begin{scope}[xshift=5cm]
% TikZ code for second picture
\end{scope}
\end{tikzpicture}

Then the points plotted at (x,y) in each scope will be exactly 5cm apart.

The code of Martin is correct but if you want to know the width and the height of the picture, you can use

\newbox\mybox 
\setbox\mybox=\hbox{\begin{tikzpicture}
...
\end{tikzpicture}} 

Now you get the width and the height of picture with

\wd\mybox
\ht\mybox
\dp\mybox

The height is \dp + \ht. I don't know if pgf/TikZ gives the height and the width directly.