From the Encyclopedia of Laser Physics and Technology:
Pump–probe measurements can be used to obtain information on ultrafast
phenomena. The general principle is the following. A sample (e.g. a
SESAM) is hit by some pump pulse, which generates some kind of
excitation (or other modification) in the sample. After an adjustable
time delay (controlled with an optical delay line), a probe pulse hits
the sample, and its transmission or reflection is measured. By
monitoring the probe signal as a function of the time delay, it is
possible to obtain information on the decay of the generated
excitation, or on other processes initiated by the pump pulses.
From the Wikipedia article on Time-resolved spectroscopy:
Transient-absorption spectroscopy is an extension of absorption
spectroscopy. Here, the absorbance at a particular wavelength or range
of wavelengths of a sample is measured as a function of time after
excitation by a flash of light. In a typical experiment, both the
light for excitation ('pump') and the light for measuring the
absorbance ('probe') are generated by a pulsed laser.
The "sample" is anything that you want to get the spectrogram of.
This article mentions photochemistry as one possible application. It also notes that:
in some applications such as spectroscopy and pump-probe experiments,
the laser wavelength must be tuned continuously during the experiment
So, it's not that pump-probe measurements are useful when dealing with tunable lasers, its that tunable lasers are useful when dealing with pump-probe measurements.