rowcolor command from colortbl package leaves unwanted gaps in shading between columns

• @{\extracolsep{\fill}} introduce additional space between columns which is not covered by rowcolor
• in second line you have \rowcolor{newgray}[20pt][20pt] which should be only \rowcolor{newgray}
• coloring rows and use rules from booktabs doesn't gives nice result, instead them i suggest lines provided by package boldline
• for more vertical space in cells i suggest to use calspace

\documentclass{article}
\usepackage[table]{xcolor} % it call and enhance colortbl
\definecolor{newgray}{RGB}{196,196,196}
\usepackage{booktabs, threeparttable}
\usepackage{tabularx}       % new
\usepackage{boldline}       % new (part of shipunov bundle)
\usepackage{cellspace}      % new
\setlength\cellspacetoplimit{4pt}
\setlength\cellspacebottomlimit{4pt}
\addparagraphcolumntypes{X}
\newcolumntype{C}{>{\centering\arraybackslash}X}
\newcolumntype{L}{>{\raggedright\arraybackslash}X}
\newcolumntype{R}{>{\raggedleft\arraybackslash}X}

\begin{document}
\begin{threeparttable}
    \centering
\begin{tabularx}{\textwidth}{ S{>{\hsize=0.4\hsize}L}
                                >{\hsize=0.2\hsize}L
                                >{\hsize=0.2\hsize}C
                                >{\hsize=0.2\hsize}R}
\multicolumn{4}{c}{\textbf{Random Table}}                               \\
    \hlineB{2}
        Item    & Case 1        &   Case 2          &   Case 3          \\
    \hlineB{1.75}
    \rowcolor{newgray}
first thing,    & \tnote{a}245  & \tnote{b}25e-6    & \tnote{c}79e-6    \\
    \rowcolor{newgray}
second thing    & \tnote{a}450  & ---               & \tnote{b}24e-6    \\
third thing     & \tnote{d}2e-6 & \tnote{c}4123.5   & \tnote{b}2e-6     \\
fourth thing    & ---           & 1e6               & no data           \\
    \hlineB{2}
    \end{tabularx}

    \begin{tablenotes}
    \item[a] some explanation
    \item[b] this is why \dots
    \item[c] see reference awesome
    \item[d] hello world
    \item
    \end{tablenotes}
\end{threeparttable}
\end{document}

edit: if you like to have simpler solution with less nice table (which width is still equal to text width), than try the following:

\documentclass{article}
\usepackage[table]{xcolor}  % it call and enhance colortbl
\definecolor{newgray}{RGB}{196,196,196}
\usepackage{booktabs, threeparttable}
\usepackage{tabularx}       % new
\newcolumntype{C}{>{\centering\arraybackslash}X}
\newcolumntype{L}{>{\raggedright\arraybackslash}X}
\newcolumntype{R}{>{\raggedleft\arraybackslash}X}

\begin{document}
\begin{threeparttable}
    \centering
    \renewcommand\arraystretch{1.2}
\begin{tabularx}{\textwidth}{ L L C R }
\multicolumn{4}{c}{\textbf{Random Table}}                               \\
    \hline
        Item    & Case 1        &   Case 2          &   Case 3          \\
    \hline
    \rowcolor{newgray}
first thing,    & \tnote{a}245  & \tnote{b}25e-6    & \tnote{c}79e-6    \\
    \rowcolor{newgray}
second thing    & \tnote{a}450  & ---               & \tnote{b}24e-6    \\
third thing     & \tnote{d}2e-6 & \tnote{c}4123.5   & \tnote{b}2e-6     \\
fourth thing    & ---           & 1e6               & no data           \\
    \hline
\end{tabularx}

\begin{tablenotes}
\item[a] some explanation
\item[b] this is why \dots
\item[c] see reference awesome
\item[d] hello world
\item
\end{tablenotes}
\end{threeparttable}
\end{document}

note:

• S{X} specifier means that to selected X column is added vertical space defined by

  \setlength\cellspacetoplimit{4pt}
  \setlength\cellspacebottomlimit{4pt}
  

• with >{\hsize=0.4\hsize} and >{\hsize=0.2\hsize} is changed ratio between columns widths (first column has width equal 0.4\textwidth, other three have width equal 0.2\textwidth each)

• the first solution can be simplified by removing cellspace stuff and for more vertical spaces around cells contents use simpler \renewcommand\arraystretch{1.2}:

  \documentclass{article}
  \usepackage[table]{xcolor} % it call and enhance colortbl
  \definecolor{newgray}{RGB}{196,196,196}
  \usepackage{booktabs, threeparttable}
  \usepackage{tabularx}       % new
  \usepackage{boldline}       % new (part of shipunov bundle)
  \newcolumntype{C}{>{\centering\arraybackslash}X}
  \newcolumntype{L}{>{\raggedright\arraybackslash}X}
  \newcolumntype{R}{>{\raggedleft\arraybackslash}X}
  
  \begin{document}
  \begin{threeparttable}
      \centering
      \renewcommand\arraystretch{1.2}
  \begin{tabularx}{\textwidth}{>{\hsize=0.4\hsize}L
                               >{\hsize=0.2\hsize}L
                               >{\hsize=0.2\hsize}C
                               >{\hsize=0.2\hsize}R}
  
   %% table body and table notes %%
  
  \end{threeparttable}
  \end{document}