Firstly you should compare like with like. If you use \showthe\linewidth on the main part of you document you will see that its value is 345.0pt which is slightly more than 12cm = 341.43306pt. The easiest way to make a genuine test is to write
\begin{minipage}{\linewidth}
...
\end{minipage}
Anyway, given that, you still get a similar difference in output. This is mostly caused by the fact that minipage calls \sloppy when it starts.
Going the opposite way to what you ask, you can undo that particular setting globally by
\makeatletter
\def\@minipagerestore{\fussy}
\makeatother
With these two changes
\documentclass[a4paper]{letter}
\makeatletter
\def\@minipagerestore{\fussy}
\makeatother
\begin{document}
\begin{minipage}{\linewidth}
This is an example paragraph inside a \textit{minipage} showing how text
should flow, notice how nothing is pushed onto the margins:
\textbf{Lorem} ipsum dolor sit amet, consectetur adipisicing elit, sed do
tempor incididunt ut et dolore magna aliqua. Ut enim ad minim veniam
quodque $P, Q , R , S ,T \ldots, X, Y, Z$, quis nostrud exercitation ullamco.
\end{minipage}
\vspace{2em}
This is an example paragraph without any sections inside the main document,
and here the math environment overflows into the margin:
\textbf{Lorem} ipsum dolor sit amet, consectetur adipisicing elit, sed do
tempor incididunt ut et dolore magna aliqua. Ut enim ad minim veniam
quodque $P, Q , R , S ,T \ldots, X, Y, Z$, quis nostrud exercitation
ullamco.
\end{document}
you get
Notice that there is no space between the paragraphs in the minipage. This is because minipage sets \parskip to 0pt, but in letter class it is otherwise set to 6.99997pt.
A list of parameters that minipage sets as standard can be found in the source2e documentation. There you can read that it also sets to 0pt the following parameters
\parindent
\@totalleftmargin
\leftskip
\rightskip
\@rightskip
Additionally, it sets
\parfillskip = 0pt plus 1fil
\lineskip = \normallineskip
\baselineskip = \normalbaselineskip
\@listdepth = \@mplistdepth
the latter is initially 0, and @minipage is set to true: quite a number of commands check this last value during their operation and so behave differently in a minipage.
Anyway, the simplest way to get minipages linebreaking is to issue \sloppy, as Steven Segletes says. Of coures, now you are throwing away many of LaTeX's attempts to do good line breaking. In general it is better to fix the bad break. In this case, you should allow breaks at the commas in your mathematics expression:
\documentclass[a4paper]{letter}
\begin{document}
\textbf{Lorem} ipsum dolor sit amet, consectetur adipisicing elit, sed
do tempor incididunt ut et dolore magna aliqua. Ut enim ad minim
veniam quodque $P,\allowbreak Q ,\allowbreak R ,\allowbreak S
,\allowbreak T,\allowbreak \ldots,\allowbreak X,\allowbreak
Y,\allowbreak Z$, quis nostrud exercitation ullamco.
\end{document}
To do this automatically see Allowing line break at ',' in inline math mode? and Break an inline math formula.
\sloppywill fix this case. – Steven B. Segletes Feb 11 '14 at 14:01\sloppyis the last option (like usage ofminipage, by the way). You can allow breaks at any place of the math (except inside groups) by\allowbreakcommand. The spaces in the first example are so huge! – yo' Feb 11 '14 at 14:44\sloppywould cause the top-level text to mimic the minipage. But you are right, the minipage is not appealing to look at. In fact, it is quite... sloppy. – Steven B. Segletes Feb 11 '14 at 14:47