When we should use only an "Analog" multimeter?
You didn't ask for a complete exposition about the differences, so I'm not going to try and tabulate anything here. You asked about where an analog meter might be better (or should be preferred.)
Probably one of the better cases to try, if you are seriously looking to see where a very high quality digital meter (such as a Fluke 87) does far more poorly than a very cheap (nearly free, by comparison) analog voltmeter (such as a TekPower TP7040 -- an inexpensive, fine unit that includes the meter mirror strip [and in my opinion is better than the TekPower TP7050]), is to set up a signal generator to provide a sine wave at \$1\:\textrm{Hz}\$ that varies from about \$3\:\textrm{V}\$ to about \$7\:\textrm{V}\$ (in short, it has a DC bias to it that you ALSO want to see.) Now, hook up both meters.
A digital voltmeter (DVM) will spend all of its time rooting around, going from ERR to who knows what, trying to "auto-range." And, in effect, pretty much NEVER telling you anything useful at all except perhaps that the signal is "difficult." Meanwhile, the cheap analog voltmeter will very nicely swing back and forth between the two values and clearly show you a LOT better detail about what is taking place. You will even have a decent idea about the minimum and maximum values and that it moves smoothly between them.
It's like night and day.
Setting a DVM to manual mode and to the appropriate DC range (when both these features are available) stops the auto-ranging behavior and allows periodic display updates of the measurement. But the values appear to be taken "at random." It's much easier to see what is going on with the analog display, for some kinds of measurements. If also available in manual mode and also with the appropriate DC range selected, setting a DVM to use still faster display update rates improves this situation, too. (My Tektronix DMM916 allows this.) But the point remains for observing some situations. Besides, all we are doing here is narrowing cases by spending more money on the DVM.
When the service manual calls out to use an analog meter (such as the venerable Simpson 260) and loading would be different using a digital meter.
Some better digital meters have an analog-like segmented LCD display that mimics a meter movement (with a relatively high sample rate) so restores some of the advantages you would see following varying signals.
You an pick up variations visually from an analog meter easily- whereas digital meters flicker in the least significant digit is just about as distracting as flicker in any other digit.
Digital meters and analog-like displays that auto-range can be even worse. Usually you can turn it off.
In some cases digital meters have very high impedance (tens of M or even G\$\Omega\$) which can lead to confusing results, whereas analog meters- those that don't contain amplifiers take a fair amount of current to crank the needle around against the torque of the hairspring.
Analog multimeters have the advantage of no battery required for measuring voltage and current. Thus they can be used in the field without worrying about batteries going bad. They are also very useful for making adjustments on circuits that require setting to a minimum or maximum. It is much easier to see such settings on an analog scale then with a digital readout. However, as far as AC voltages go, most analog meters are calibrated to read the RMS value of a sine wave and will be inaccurate with other waveforms. Many digital meters, however, have RMS to DC converters, and will read the correct RMS value for sine waves and most other AC waveforms including triangles and square.