To create a new environment, it has to have the form
\newenvironment{<env>}
{<begin env>}% \begin{<env>}
{<end env>}% \end{<env>}
where <begin env> is execute at \begin{<env>} and <end env> is executed at \end{<env>}. Using your approach causes confusion, even though you think it might work:
\newenvironment{note}
{ \footnote{ }
{ } }
For example, even though you think \begin{note} should expand to \footnote{ and \end{note} should expand to }, thereby embracing the enviroment contents, TeX creates the environment at definition time, and the braces in your definition is ambiguous. In fact \begin{note} actually expands to \footnote{ } { } since the balanced pair of braces are matched at the same level/depth.
Your way around this predicament is to capture the environment contents first - made possible with the aid of environ, and then pass that to \footnote. The environment contents is grabbed inside a macro called \BODY:

\documentclass{article}
\usepackage{environ}% http://ctan.org/pkg/environ
\NewEnviron{note}{\footnote{\BODY}}% Environment-form of \footnote
\begin{document}
Hello\footnote{world}. Hello\begin{note}world\end{note}.
\end{document}
Note that, even with the environment-like approach, you're still limited in terms of what can be contained within (no verbatim content, for example).
\pfootnotecommand that you can redefine at will? – egreg Nov 09 '13 at 11:23