Fourier transforms are very important for a variety of spectrocopic techniques, for example NMR and infrared spectroscopy. Before computers were suffiecently powerful to handle the algorithm quickly enough, all measurements were done in a continuous wave mode. The difference between the two methods FT and CW, can be illustrated as follows :-
Say you've made a church bell, it's overall tone will be made up of a variety of frequencies and harmonics; how good the bell sounds depends of the exact amplitude of each frequency produced. You could quantify all this by making measurements of the bell's performance....
One way to do this would be to take a frequency generator, loudspeaker and a microphone. You would put the speaker and the microphone inside the bell, and then generate discrete frequencies across the whole range you might expect to get a response and play them through the speaker. You use the microphone to measure the bells response at each frequency, plotting the amplitude response on graph paper as you go. This would give you a spectrum, characteristic of the bell. Note to make very fine measurements, you have to sweep very carefully, like trying to tune into one of two very close FM radio statons, which takes time. Also the amplitude of the signals may be very weak, so sensitive equipment may be needed.
A far better way is to just take the microphone, stick it in the bell, and whack the bell with a hammer! Now the output will be a very complicated waveform, consisting of all the frequencies the bell can produce, decaying over time. The fourier transform can take this time based information and convert it back to the frequency domain to give the same result as above.
The free inductive decay produced is commonly digitized and stored in a computer. This method has the advantages of being very quick, easy to add the results of many experiments, (to let the noise average out to near zero), and easy to perform signal enhancements on the raw data. Also the resolution (ability to distinguish to close together frequencies) is largely a product of how good a digitizer you can make. To give you an example an NMR spectrometer aquiring data in continuous wave mode might take ten minutes to record a very high resolution proton spectrum that consists of just one scan. In fourier transform mode it can do one scan every two seconds (or less)! The same result can therefore be obtained in a fraction of the time, or the signal to noise quality much improved in equal time.
When FT spectrometers first came on the market, they gave a huge boost to science, as they made what was very difficult, routine.