This is a bit old, but here are some things to think about.
First, the title is a bit misleading. The title is "time-frequency domain" which is, really, something else. (more on that later).
The whole point of representing a signal in the frequency domain is to be able to see things with more clarity, and to gain more info. The frequency domain also has amplitude.
Fourier transform is about decomposing a signal into sinusoïd of different frequencies. Each sinusoïd with a different frequency 'contributes' its own way so that the 'whole' gives you back your original signal.
This is useful in two ways: Deconstruction, and reconstruction.
Deconstruction is obvious, you want to see which frequencies are in a signal.
Reconstruction is something you don't think about on the spot (as it is presented in courses): It allows you to go the other way. You can "build" a non sinusoidal signal using simple sine waves of different frequencies.
So for example: You can build a square wave using sine waves, which isn't something one thinks about, really.
Another thing is bandwidth. By decomposing a signal, you answer the question: Which frequencies bear the most power or information.
So if the signal you break down is concentrated in the first three harmonics, you don't need to transmit the other ones and save a lot on bandwidth. So instead of transmitting the whole signal, you transmit just the frequencies that are the most relevant.
Also, it enables you to see clearly.
For example, let's say you have a signal that is modulated by another (let's assume they're both sine waves). In the time domain (i.e amplitude vs. time), the result of such modulation is a mess, but in the frequency domain it's so clear, it jumps at you: The frequencies present in the signal are represented by delta functions. You couldn't tell in the time domain.
And finally, I said earlier that the title is misleading.
There is time representation. There is frequency representation.
There is also time-frequency representation. That is, the signal is both represented in the time-domain and in the frequency domain.
This was needed because Fourier transform tells you about the spectral characteristics of a signal, but doesn't tell you how these change over time.
In other words: It tells you that this frequency exists in that signal, but it doesn't tell you when it appears. Just that it's there.
This is why there's what is called TFRs (Time-Frequency Representations) Joint Distributions, etc..
Have a look at Wigner-Ville Distribution, look up Leon Cohen, too.
If you want to dig deeper, take a look at "Time Frequency Signal Analysis and Processing, A Comprehensive Reference". Elsevier. Edited by Boualem Boashash.
There's also the excellent "Time Frequency Distributions, A Review" by Leon Cohen.