Not sure if it was clear to everyone that the question was not about pentodes vs. triodes, but was about pentodes operated as pentodes vs. pentodes operated as triodes.
The initial answer by John (Jmcgrogan2) pretty much said it all in a nutshell, IMO. It is common for triode mode operation to sacrifice something like 40% or 50% of the power capability of pentode mode operation, while providing some improvement in sound quality, mainly by providing an approximation of the greater linearity that a pure triode can provide in comparison to a pentode or other amplification device (as Atmasphere indicated).
See this
Wikipedia writeup for a technical overview. Note the section toward the bottom headed "Triode-strapped pentode circuits." Basically, connecting the middle of the three grids of a pentode (the "screen grid") to the plate, usually through a low value resistor, causes the tube to exhibit triode-like characteristics. A triode being a tube having only one grid, corresponding to the "control grid" shown in the article.
The amplifiers I have had that could be switched between the two modes were two pairs of the classic vintage Marantz monoblocks, the Model 2 and the Model 9, which I owned during the 1990's. Using them in conjunction with 90 db speakers having easy to drive impedance characteristics, in both cases triode mode provided a considerably richer and more natural sound that was clearly preferable to pentode mode, as long as I didn't run out of power. Which in turn only happened on a limited number of recordings having exceptionally wide dynamic range, such as some of the Telarc symphonic recordings with their notorious bass drum beats.
Regards,
-- Al