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I want to use the Palatino font without changing the math font, and it seems to me that both \usepackage{palatino} and \usepackage{newpxtext} give the same result.

Can I use either?

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You can of course use either, but use newpxtext

The psnfss collection was designed to use the "real" Adobe fonts available in the new laser printers becoming available, palatino.sty is just

\renewcommand{\rmdefault}{ppl}
\renewcommand{\sfdefault}{phv}
\renewcommand{\ttdefault}{pcr}

that is it sets palatino for roman, helvetica for sans and courier for monospaced, with no other adjustments.

With modern requirements to embed fonts in the PDF rather than rely on fonts built in to a printer, in fact by default clone fonts will be used so as you comment the end result is similar to newpxtext but newpxtext is far more extensive using TeXGyre Pagella and with several options to adjust commands to better use that font set and using npxss for sans serif which is again based on Helvtica but adjusted to patch the roman font, and similarly npxtt for monospace.

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