Question: Could someone explain to me please which applications demand one or the other and why? As far as I have read it's all about the 'dB'; is that true? And why?

At first I can see Digital Storage Oscilloscopes (DSO) with FFT function and Spectrum Analyzers (SA) as being the same thing...they will get a signal from the Time Domain, and convert it to the Frequency Domain and we can check all the harmonics and frequency components of a signal and analyze it in a whole new way.......But since DSOs usually are much cheaper than SA, I keep wondering what functionalities the SA will offer that a DSO can't. Is it about precision, speed of calculation (my DSO FFT is really slow), bandwidth (cheap DSOs usually go only up to 100MHz), or does it just depend on the models and not on being a DSO or a SA? Is there more that I don't know about and you can tell me?

Answer: 
To answer simply - an oscilloscope is an essential tool for any electronics lab, whilst an SA is generally not (unless you are an RF engineer, and even then you need a good scope) and for a good quality one much more expensive in comparison (though Rigol have just brought out some pretty powerful SAs at decent scope type prices)  

The FFT function on your average DSO will do for most work, so unless your frequency range of interest is e.g. > 500MHz or so (if it is let us know), then the DSO is the tool of choice. 

Basically one does amplitude versus time (scope), and the other does amplitude versus frequency (SA) 

Scope example:  

Say you have a digital signal that is intermittently working, you could check on the scope and look for over/undershoot, ringing, noise, gltiches, etc. 

![Integrity problems](https://i.stack.imgur.com/1HIa6.gif)

(simple) SA example:
Say you have a signal and you want to check the harmonic components of it, you can look on the SA screen and check for harmonics (e.g. a pure sine wave should just be one single spike on the screen, at it's frequency, a square wave would be a decreasing series of odd harmonics) 

Square wave on a Spectrum Analyser: 

![SA Square wave](https://i.stack.imgur.com/5Whlb.png)

The same signal on a scope would look like this:

![Square wave on scope](https://i.stack.imgur.com/I8AHu.gif)
The difference is that the spectrum analyzer has a mixer frontend allowing it to shift the frequency range it is listening to, while an oscilloscope remains fixed at the lower end.

This means that it is possible to see signals at higher frequencies, and at the same time, signals outside of the area being looked at are filtered out, so you can adjust the ADC prescaler for a better resolution.

On the other hand, mixers don't like DC at all, so in normal EE work, you won't be able to use a spectrum analyzer in place of an oscilloscope either.