Peak height is routinely used when integrating the peak on a downslope.  This is common in GC particularly when the peak is close to the injection front.  It works as long as you can demonstrate linearity or a correlation in peak response versus concentration.

Peak height can be used in some cases, but is not the "norm" or acceptable in most methods. For HPLC analysis, area is the preferred measurement technique as it does a better job with peak shapes that are not perfectly symmetrical. Peak height alone can be very accurate IF the baseline noise is low, the sample is pure, no matrix interference and you have a true Gaussian shaped peak (close to perfect, which may be the case in well developed cap GC methods, but almost never in HPLC).

• For HPLC methods, please use the far more accurate and reliable 'Peak Area' for measurements.
• Background Info: A very long time ago, when HPLC was still young and in development, GC was far more commonly used. Peak 'amounts' were determined by peak height alone as all of the signal capture methods used at that time were very basic in design (mostly strip chart recorders). No reasonably priced commercially available computer integration devices or methods were available to us and measuring the amplitude of a peak (in voltage) was easy. That is why so many early methods were published with peak height (peak amplitude) only. To calculate peak areas, we would take a pair of scissors and cut out the peaks from the strip chart paper, then weigh them on an analytical balance. This allowed us to measure peak area as a percentage and it worked rather well (it still works well and I have even demonstrated the technique to scientists working in remote areas who only had primitive instruments to use). Once computerized integrators were introduced (thank you Hewlett-Packard!), no more cutting of paper peaks was needed as the integrators were able to use calculus to measure the area under each curve quickly and accurately, then provide area % data which was far more accurate than peak height data for real-life samples.

I think when we got a narrow band width

You can decide for your specific application which is better by performing several standard additions into the sample and check the correlation and linearity of peak area versus peak height.

Use of peak height was common when you used strip chart recorders because it was much more convenient to measure the height than to measure the area of the peak. Assuming that you are using a computerized system there is never a reason to use peak height over peak area; peak height measurements make assumptions about your peak geometry that are not made in peak area measurements.

Peak height is mostly used while setting the limit  of quantification and detection (LOD & LOQ) values. It also depends on the application of your analysis, if it is bound to follow some rules and regulations then you should follow the guide lines provided by the authority. And if it is not bound to follow any rules and regulations then you can inject number of standards and can check the linear response over the range of concentration for both the area and peak height & then you should use the one which shows the better linear response (regression coefficient or correlation coefficient). Develop your own method and validate the method (if required).

I never use peak height. Your chromatography has to be perfect and consistent to use peak height otherwise peak broadening could become an issue.

If the response of your detector isn't quite linear, this will have more effect on peak height than peak area, because the aberration affects the top of the peak more than the sides. With UV-vis absorbance detectors the specific absorbance of your analyte is likely to be independent of concentration within the working range. However, because of necessary compromises in instrument design, the detector response may begin to decline significantly at absorbances of 1AU or even a bit lower.

Non-linearity is less predictable with mass spectrometers.

Having said that, during the early days of GC-MS (I don't have the reference), someone demonstrated that when baseline noise is significant, peak height measurement is more reliable than integration. One possible explanation is that software was not as good as pencil and paper for dealing with baseline noise, a situation that may still apply. While this isn't relevant to most regulatory analytical work, where the S/N ratio must be high, in other fields we do sometimes have to work uncomfortably close to the detection limit. It should be said, though, that software should provide means of combining data from repeat injections.

Finally, I agree with Michael that it's necessary to consider any drift in chromatographic parameters during a run sequence. One consideration is drift in the effective flow rate due to temperature drift at the pump; remember, we integrate with respect to time not volume.

In HPLC analysis you cannot go with peak height as some peaks may be broad and some may be having narrow peak width, thus broader peaks will have less height as compared to narrow peak, though you may have same concentration for both peaks component. The method you are talking from USP may be old and traditional methods.

Lloyd Snyder did a great study on the effects of peak separation and peak broadening with peaks. The answer to your question depends greatly on the degree of separation and the relative sizes of the peaks.  The choice of baseline, peak detection and smoothing parameters are very important, too. See Jack Kirkland and Lloyd Snyder's Intro to HPLC for the details. I use the DAQ2GO smoothing feaytures and the ability to separately define peak start, peak stop, baseline start and baseline stop independently. See https://www.youtube.com/channel/UCYckp_D-1czwBwmQmRp4cPA

I used both approaches (peak height and peak area) many years ago in graduate school for HPLC and GC analyses.  As long as the sample and standard showed very similar peak shapes, the results were very similar between the two approaches.  However, if the samples contained a different matrix (or different solvent), then the peak shape variation between the standards and samples could vary too much to use peak height.  Peak integration is easy to obtain on the instruments used today, so I would suggest using this approach unless you only have a strip chart recorder available.  Even with a strip chart recorder, a planimeter can be used to obtain peak areas as an alternative to cutting out peaks and weighing which can be very time consuming and tedious.

We have used data files and evaluated the differences in Excel using the DAQ2GO(R) data treatment workbook.  We have also evaluated the impact of various smoothing effects on twhich is better. 

One important thing to realize is that peaks of supposedly the same material in a standard and a sample need to have the same peak shape to be considered 'real' for the sake of comparisons, calibrations, and quantitation. When the peaks are of different shape, that is an indication of imperfect separation, and that means some materials are co-eluting. The only common exception to this assertion is in the cases of column overloading, in which case the chromatography needs to be improved.

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Scott is right in stating that peak shape differences can be the result of a co-eluting impurity.  However, when large amounts of plant or animal matrices are present, the change in peak shape is often the result of interaction(s) with the matrix.  If you are dealing with relatively clean water samples, the peak shapes and retention times should be consistent unless you are dealing with ionic compounds where pH of the samples can have an impact as well.

If you are determining LOQ by signal to noise ratio, at that time peak height is important ...LOD should be determined by S/N ratio, not other methods..

I agree with Bill Letter, and would add to that by saying that no reference method will ever use peak height; all reference methods that I am aware of use peak area (unless the reference method pre-dates electronic integration).There really is no reason at all to use peak height.