What's the difference between frequency domain and time domain spectra? 
If I have a mechanical oscillator and want to observe the dynamical behavior of the oscillator, is there any additional information to observe it in time domain and frequency domain? Normally, we observe the frequency domain spectra (power spectral density) as the information of oscillator. In fact, I'm solving a dynamical behavior of two coupled mechanical oscillator, like the picture above. While someone told me that I could get different information from time domain than frequency domain. In my opinion, the difference between time domain and frequency domain is just the transform of Fourier. So what's the difference
 A: You are correct that time and frequency domains are just Fourier transforms of each other. However you only have full information if you have amplitude and phase information (as opposed to a power spectral density which is only amplitude information in the frequency domain).
Frequency spectra might tell you that you have multiple modes that exist, but it won't tell you what the phase is between modes or even if the modes are coherent or not (i.e. whether there is any definite phase between modes at all). As an example the PSD for the image you give will show that you have two modes (assuming the modes are not degenerate), but you won't be able to tell if you have two coupled oscillators (like you show) or two completely independent modes (not coupled). 
A: A single oscillator will make the rope tied between the two walls, to display standing waves, i.e. any point of the rope will rise and get down according to a law 
f(t) = Asin(ωt), where A is the maximal amplitude of oscillation at that point.
But you have two oscillators. The oscillation of a point in time is
(1) f(t) = A_1 sin(ω_1 t) + A_2 sin(ω_2 t).
Now, if someone knows nothing about the presence of the two oscillators and it is just given the graph of f(t) (not the formula) the graph may not look simple. But if he does the Fourier transform he will obtain a function looking like
(2) F(ω) ~ A_1 [δ(ω - ω_1) - δ(ω + ω_1)] + A_2 [δ(ω - ω_2) - δ(ω + ω_2)]
where δ is δ Dirac.
Now, he will know that he has two oscillators, and will be able to infer the formula (1) and other data (energy of oscillation, etc.)
Good luck
A: A problem formulated in the time domain and its equivalent formulation in the frequency domain contain essentially the same information. They just have different mathematical forms. One is easier to solve than the other, that is why we use transformations.  
Many problems attempted through the equations of motion obtained from Newton's equations are really difficult, because they are intrinsically in the time domain and the quantities involved are forces which are vectors. 
A: A signal spectrum can be decomposed in its component frequencies in two ways:


*

*Time domain - For example with wavelets (you will see the different frequencies along the Y axis and the increasing time in the X axis)

*Frequency Domain - For example with the Fast Fourier transformation or multitaper transformation where you will find the frequency power in the Y range and the frequency of time in the X axis. Usually the time frequency has as its maximum range half the length of the total time and is given in percentage of the range. 
A: There are no differences. Frequency domain is used as a mathematics transformation tool (Fourier, Laplace) in order to resolve too complex differential equations in time domain spectra.
