LuaLaTeX: \math... (\mathtt, \mathcal, \mathfrak) vs. \sym... (\symtt, \symcal, \symfrak) commands and setting different fonts

Question

In another answer (for a completely different question) I have been advised to use the \sym...-commands to change the font in math mode instead of the traditional \math...-font commands, if I use LuaLaTeX and unicode-math.

Question 1: Why? I tried to read the fontspec documentation but did not get the point. Especially, I wonder when should I use the one or the other alternative? I believe there are probably valid use cases for the traditional \math...-font commands. Otherwise, those commands could have been redefined to be equivalent to the new \sym...-commands.

I use Libertinus for my project and redefine some fonts, because I do not like the default selection which are made by the Libertinus package. However, I made some strange observations.

The MWE with nothing else than the pure Libertinus package:

\documentclass[a4paper]{article}

\usepackage{libertinus}

\begin{document}
\begin{tabular}{lcc}\hline
                  &  \verb#\math...#      &  \verb#\sym...#      \\\hline
\verb#...frak#    &  $\mathfrak{F}$       &  $\symfrak{F}$       \\
\verb#...cal#     &  $\mathcal{C}$        &  $\symcal{C}$        \\
\verb#...tt#      &  $\mathtt{Teletype}$  &  $\symtt{Teletype}$  \\\hline
\end{tabular}
\end{document}

Question 2: Both fraktur commands (\mathfrak, \symfrak) do not seem to change the font at all. Why? What is wrong?

Question 3: The fraktur and calligraphic fonts do no seem to differ between the \math...-variant and the \sym...-variant. This is somehow expected. But the oldstyle \mathtt does not seem to select a teletype font. Maybe it is mono-spaced (it is hard to tell), but it still has serifs. Why? Also it obviously differs from the \symtt-font, which I would call a real teletype font. Why do \mathtt and \symtt differ?

I tried to select a different fraktur, calligraphic and teletype font. See this MWE:

\documentclass[a4paper]{article}

\usepackage{libertinus}
\setmonofont{Inconsolatazi4-Regular.otf}[Scale=MatchLowercase]
\setmathtt{Inconsolatazi4-Regular.otf}[Scale=MatchLowercase]
\usepackage[cal=pxtx,frak=pxtx]{mathalfa}

\begin{document}
\begin{tabular}{lcc}\hline
                  &  \verb#\math...#      &  \verb#\sym...#      \\\hline
\verb#...frak#    &  $\mathfrak{F}$       &  $\symfrak{F}$       \\
\verb#...cal#     &  $\mathcal{C}$        &  $\symcal{C}$        \\
\verb#...tt#      &  $\mathtt{Teletype}$  &  $\symtt{Teletype}$  \\\hline
\end{tabular}
\end{document}

The fonts are as I want them, but only if I use the old-fashioned \math...-commands. Also, the fraktur font is a real fraktur font. But obviously, the \setmathtt-command from the fontspec package and the mathaplpha package only affect the traditional commands for font selection in math mode.

Question 4: If I am supposed to use the new \sym...-commands (for whatever reason), how to I change the font of the \sym...-commands according to my needs?

Answer 1

To answer question 1: All (or at least most) \mathXX are internally the same a, they only look different as they are from different fonts. The sym-variants are different symbols with different unicode code points. You can see the difference if you try to copy and paste from a pdf.

\documentclass{article}
\usepackage{unicode-math}
\begin{document}
$\mathrm{a}\mathtt{a}\mathsf{a}\mathbf{a}$

$\symrm{a}\symtt{a}\symsf{a}\symbf{a}$
\end{document}

This copies and pastes like this:

aaaa
a𝚊𝖺𝐚

The difference matters a lot for accessibility.