PGF Keys differences between .initial and .default

They're for two different things: A key defined using /.initial=<value or string> is a value-storing key, with the initial value set to <value or string>.

The /.default keyword defines what value will be used as the argument for a key defined with /.code=<code> if no argument is provided.

The two things are similar, but not equivalent. A key can at the same time have a value, which can be queried with \pgfkeysvalueof{/key}, and a code, which will be run when you just call the key using \pgfkeys{/key}.

If no .code is defined, you could use either .initial or .default with pretty much the same behaviour: you set the value using \pgfkeys{/key/.initial=value}or \pgfkeys{/key/.default=value}, and query the value using \pgfkeys{/key}. However, if you have defined a .code, as well as a value using .initial, to query the value you need to call \pgfkeysvalueof{/key}.

\documentclass{article}
\usepackage{pgfkeys}

\begin{document}
%% Case 1: We want a key to store a value

% Just passing a value to an undefined key fails:
%\pgfkeys{firstkey=red}

% It first needs to be initialised
\pgfkeys{firstkey/.initial=red}

% Then we can get the value
\pgfkeys{firstkey}

% After the key has been initialised, we can change the stored value using simple assignments
\pgfkeys{firstkey=blue}
\pgfkeys{firstkey}

%% Case 2: We want a key to execute code using the argument
\pgfkeys{secondkey/.code=Your argument: \textbf{#1}}
\pgfkeys{secondkey=Some words}

% If we don't use an argument, it's assumed to be empty
\pgfkeys{secondkey}

% We can provide a default value to be used if no argument is provided:
\pgfkeys{secondkey/.default=Nothing}
\pgfkeys{secondkey}

% Let's define a .code key that just returns the argument...
\pgfkeys{/thirdkey/.code=#1}
% ...and give it an initial value...
\pgfkeys{/thirdkey/.initial=green}
% ...and a default argument
\pgfkeys{/thirdkey/.default=red}

% If we query the value, we get "green"
\pgfkeysvalueof{/thirdkey}
% If we run the code, we get "red"
\pgfkeys{/thirdkey}

\end{document}

Although this does not specifically add anything to the answer already provided, the following example helped me understand the difference more intuitively:

\documentclass{article}

\usepackage{pgfkeys}

\setlength{\parindent}{0mm}

\begin{document}

%% Create the keys, default values, etc.
\pgfkeys{%
  /mythingy/.is family, /mythingy,
  usecolor/.default = green,
  usecolor/.code = {You have chosen to use the color #1.},
  usecolor/.initial = purple,
}

%% Create a command that will use the keys
\newcommand{\mythingy}[1][]{%
  \pgfkeys{/mythingy, #1}   %% The usecolor/.code is executed
  \pgfkeysvalueof{/mythingy/usecolor}   %% Retrieves only the VALUE of the usecolor key
}

%% Use the command along with the keys in various different manners

\textbf{Attempt 1}:\\
%% Here, the usecolor/.code is executed using "blue" as the argument
\mythingy[usecolor=blue]

\vspace{\baselineskip}

\textbf{Attempt 2}:\\
%% Here, the usecolor/.code is execute using the /.default argument value "green"
\mythingy[usecolor]

\vspace{\baselineskip}

\textbf{Attempt 3}:\\
%% Here, the usecolor key is initialized to the /.initial value "purple"
%% However, the usecolor/.code is not executed !!!
\mythingy

\end{document}

which outputs to:

In understanding this behavior, it is very important to understand the conceptual difference between the VALUE of a key and an ARGUMENT for the code associated to that key.

I could imagine this distinction being used in the following manner (see code below), although this example is grotesque (in my opinion) in the sense that it should use a distinct second key radius to actually control the radius:

\documentclass{article}

\usepackage{tikz}
\usepackage{etoolbox}

\setlength{\parindent}{0mm}

\begin{document}

\pgfkeys{%
  /mypainter/.is family, /mypainter,
  drawcircle/.default = true,
  drawcircle/.code = {%
    \ifstrequal{#1}{true}{
      \begin{tikzpicture}
        \draw (0,0) circle[radius=\pgfkeysvalueof{/mypainter/drawcircle}];
      \end{tikzpicture}
    }{}
  },
  drawcircle/.initial = 1cm,
}

\newcommand{\mypainter}[1][]{%
  Executing {\ttfamily drawcircle/.code}: \pgfkeys{/mypainter, #1}
  \par
  Value of key {\ttfamily drawcircle}: \pgfkeysvalueof{/mypainter/drawcircle}
}

\textbf{Attempt 1}:\\
\mypainter[drawcircle=true]

\vspace{3\baselineskip}

\textbf{Attempt 2}:\\
\mypainter[drawcircle]

\vspace{3\baselineskip}

\textbf{Attempt 3}:\\
%% The order of the "keys" is important here (try switching the two)
\mypainter[drawcircle/.initial=2cm, drawcircle]

\vspace{3\baselineskip}

\textbf{Attempt 4}:\\
%% Notice the drawcircle/.initial now remains set to 2cm !!!
\mypainter

\vspace{3\baselineskip}

\textbf{Attempt 5}:\\
\mypainter[drawcircle=false]

\end{document}

The output to that last bit of code is: