In what case do we need to increase the length of the signal compared to zero padding when using FFT?
Zero-padding before an FFT results in a form of high quality (Sinc kernel) interpolation of the resulting upsampled spectrum.

The interpolation of frequency peaks works well when the level of the surrounding spectrum, both the noise floor and any adjacent frequency energy, is low enough to not distort the interpolated peak.  (The examples shown in the question have a fairly high S/N).

But an interpolated spectral peak is very rounded after high ratios of zero padding (the result of convolution with the added rectangular window caused by zero padding).  Thus, any surrounding spectral energy can change the shape of this wider spectral hump enough to change the location of its peak in relation to the frequency of any actual underlying sinusoidal or other narrow band energy.  Two peaks close together can either get spread apart or end up appearing to be just one wide spectral peak in the upsampled interpolated FFT spectrum.  This problem becomes even worse when trying to find or measure a spectral notch by interpolation.

So if you are looking for more information about a signal with a relatively high local S/N (signal-to-adjacent-noise ratio) around the signals of interest, zero-padding before an FFT may be helpful (providing a computationally efficient wide band interpolator).

If you are looking for more information about spectrum just above the noise floor, or looking for information about closely spaced sinusoids or spectral peaks or notches, then more samples will be needed to avoid distortions due to interpolation of too little information about the spectrum caused by just zero-padding an FFT.

Also, the Sinc interpolation done by zero-padding a DFT of strictly real data is actually a dual Dirichlet (periodic Sinc) kernel interpolation, which causes even greater distortion in the interpolated spectrum near the DC and Fs/2 bins, or when used with very short amounts of actual data in relation to the size of the FFT.  More actual data is likely to be needed to get a more accurate and detailed spectrum near the DC or Fs/2 result bins of a DFT.