"ProRes" and "FastRes" are HP terms and not general terms. Nobody but HP knows what's involved in one or the other. Print quality is a very subjective area - things like paper quality and cleaning matter a lot. Some printing methods are more suitable for certain paper types than others.
LPI stands for lines per inch, so generally the higher, the better. Same for DPI, dots per inch. Normally printers express quality in DPI and if they use some other unit, I'd suspect they are up to something fishy, trying to make something sound better than it actually is.
This question on Super User is literally a duplicate of your question. To quote the accepted answer from there, by the user Damon:
FastRes1200 is 600 DPI with HP's special secret sauce added, i.e. variable dot size. This is in reality still 600 DPI, which however simulates 1200 DPI. According to HP, the quality is near-identical or superior to true 1200 DPI (to be honest, I can't tell a difference, either -- looks just fine to me).
Advantage: 1/4 as many pixels to render as with true 1200 DPI, thus much faster. In fact, it's exactly the same as if you used the 600 DPI setting, only just now the printer actually puts on the physical page whatever greyscales it has rendered (instead of thresholding and discarding that information!).
ProRes1200 is true 1200 DPI according to HP with constant point size, they recommend it for printing documents with minute details such as technical drawings with fine lines.
To be honest, I have my doubts insofar as 132 LPIs (or 180 LPIs) very clearly states that the resolution is not 1200x1200, but rather 1200x132 or 1200x180, respectively.