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)
There were a few correct differences mentioned above, I will try to systemize:

1) Bandwidth (oscilloscope's bandwidth usually wider, but the working band cannot be shifted). I.e. for example Oscilloscope modes are: 0-1kHz, 0-10kHz, 0-50kHz, 0-250kHz, 0-500kHz, 0-2MHz, 0-20MHz,0-100MHz signals, having max sample rate at 500 MSamp/sec. When one looks at FFT, he can see only these 0-100 MHz band.
Spectrum Analyzer may have narrower bandwidth, but it can roll across frequency scale: i.e. for example, bandwidth 40 MHz, sampling frequency 200 MSamp/sec, and working frequencies: 0-6.3 GHz. I.e. Spectrum analyzers modes will be: 0-40MHz,10-50MHz, 20-60MHz, 30-70MHz....6260..6300MHz.
So one can see, that SA has a tunable band filter instead of anti-aliasing LPF in the oscilloscope.

2) Dynamic range. ADC of a spectrum analyzer has a much better resolution.

3) The spectrum analyzer has a low-noise amplifier, oscilloscope doesn't have it. Low noise amplifier, is a special, radio-frequency amplifier, which works in a big range of frequencies, add very low noise to signal.

4) Oscilloscope and spectrum analyzer have different ways to set up triggers. The oscilloscope is oriented on a shape of signal in the time domain, SA is oriented on capturing certain shapes in the frequency domain.

5) The oscilloscope cannot demodulate signals, a spectrum analyzer usually can (because it is virtually an SDR-receiver).

Summarizing: 
an oscilloscope is an extra-wide band millivoltmeter.
spectrum analyzer is a pretty narrow-band receiver, whos main goal to convert radio-waves into the baseband signal (I and Q components) with as low as possible lose and noise.